On the nature of magnetization states minimizing the micromagnetic free energy functional.

Università degli studi di Napoli “Parthenope”

Dipartimento per le Tecnologie

Corso di dottorato in Ingegneria dell’Informazione

xxv ciclo

On the nature of magnetization states

minimizing the micromagnetic free energy functional

Anno: 2012

Autore: Giovanni Di Fratta

Tutor: Prof. Massimiliano d’Aquino

Coordinatore: Prof. Antonio Napolitano

Table of contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1. Physical motivations: Weiss-Heisenberg theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2. Technological Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1. The modern recording process: hard disks and MRAMs. . . . . . . . . . . . . . . . . . . . . . 9

1.3. Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2. Basic Magnetostatic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1. The Lorentz force and the magnetic induction field B. . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2. The fundamental equations of magnetostatics in free space. . . . . . . . . . . . . . . . . . . . . . 15

2.2.1. Laplace’s first formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2. Gauss’s law for magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3. Ampère’s circuital law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.4. Ampére equivalence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.5. A first look to the demagnetizing factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3. Magnetized Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1. The fundamental equations of magnetostatics in matter . . . . . . . . . . . . . . . . . . . . 19

2.3.2. Researching a constitutive relation: Lorentz and Weiss ideas, Micromagnetics. . . . . . . 20

2.4. Classical aspects of atomic magnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1. The angular momentum µL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2. The spin momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5. The magnetization vector and its relation with microscopic currents. . . . . . . . . . . . . . . . 25

2.6. The constitutive relation BL=L(M) between the local field and the magnetization. . . . . . 26

2.6.1. The Lorentz local field: the Lorentz sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6.2. The Weiss molecular field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7. The constitutive relation M=F(BL,B) between the local-field Hl and the macroscopic field M and

B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7.1. Larmor precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7.2. Magnetization by orientation. Langevin function. . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8. Diamagnetism, Paramagnetism and Ferromagnetism: a microscopic interpretation. . . . . . . 31

2.8.1. Diamagnetic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8.2. Paramagnetic materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8.3. Ferromagnetic materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3. Volume and Surface Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1. The Laplace Operator and the Poisson’s Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.1. Green’s identities for bounded domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.2. Boundary value problems: uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2. The Stokes identity on bounded and regular domains. . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1. The Newtonian potential. The simple- and double-layer potentials. . . . . . . . . . . . . . 40

3

3.3. The Stoke identity on unbounded domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1. Maximum principles on exterior domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4. Surface potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4. The Demagnetizing Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1. The Newtonian Potential. Regularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2. The Helmholtz-Hodge decomposition formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1. The magnetostatic scalar and vector potentials. Integral representations. . . . . . . . . . 53

4.2.2. Transmission conditions for the magnetic flux density field b and the demagnetizing field h.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3. The L2 theory of the demagnetizing field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5. The Demagnetizing Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2. Main result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3. The Homogeneous Ellipsoid Problem and the Demagnetizing Factors. . . . . . . . . . . . . . . . 64

6. Micromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1. The general problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1.1. Forces involved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1.2. The variational approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2. Thermodynamic relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2.1. The internal energy state function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2.2. The First law of Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2.3. The second law of thermodynamics: irreversible transformations. . . . . . . . . . . . . . . 72

6.2.4. Thermodynamic potentials for magnetic media. . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3. Free-Energy Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.3.1. The magnetostatic self-energy term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3.2. The Anisotropy energy term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3.3. The Exchange energy term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4. The Gibbs-Landau free energy functional GL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.4.1. The Gibbs-Landau free energy functional GL in normalized form. . . . . . . . . . . . . . . 81

7. Equilibria of GL. Brown’s Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.1. The existence of minimizers for GL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.2. A first glance to the local equilibria of GL. First order (external) variation of GL. . . . . . . . 84

7.2.1. Weak Euler-Lagrange equation for GL: weak Brown’s static equation. . . . . . . . . . . . . 85

7.2.2. The regular case: Brown’s static equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.2.3. Brown’s static equations for uniform magnetizations. . . . . . . . . . . . . . . . . . . . . . . 87

7.3. A first glance to the local minimizers of GL. Second order (external) variation of GL. . . . . . 87

8. Global Minimizers of GL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.2. Formal theory of micromagnetic equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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8.3. The magnetostatic self-energy. Mathematical properties of the dipolar magnetic field. The Brown

lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.4. The case of ellipsoidal geometry. Demagnetizing tensor . . . . . . . . . . . . . . . . . . . . . . . . 93

8.5. The exchange energy and the Poincaré inequality. Null average micromagnetic minimizers . 94

8.6. The generalization of the fundamental theorem of Brown to the case of ellipsoidal particles . 96

8.7. Some remarks on the value of the critical size. The best Poincaré constant in the case of a spherical

particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.8. Final considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9. Local Minimizers of GL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9.1. Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9.1.1. Locally minimizing p-harmonic maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9.1.2. Some useful result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

9.2. A general stability/rigidity result. Proof of Theorems 9.4 and 9.5 . . . . . . . . . . . . . . . . . 105

9.2.1. Proof of Theorem 9.5 (Regularity properties of the Micromagnetic Energy) . . . . . . . . 107

9.2.2. Proof of Theorem 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

9.3. Proof of Theorem 9.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

9.3.1. Domain dilations (proof of parts i, ii of Theorem9.6 and preliminaries for iii) . . . . . . . 113

9.3.2. Domain translations (proof of theorem 9.6.iii) . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9.4. Proof of Theorem 9.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9.4.1. Proof of Theorem 9.9.i (inner variations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9.4.2. Proof of Theorem 9.9.ii (target variations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

9.5. Concluding remarks and further generalizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

9.6. Appendix A (proof of Proposition 9.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

10. Composite Ferromagnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

10.1.1. The Landau-Lifshitz micromagnetic theory of single-crystal ferromagnetic materials . . 134

10.1.2. The Gibbs-Landau energy functional associate to composite ferromagnetic materials . 135

10.1.3. Statement of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

10.2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

10.2.1. Γ-convergence of a family of functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

10.2.2. Two-scale convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

10.3. The Homogenized Gibbs-Landau Free Energy Functional . . . . . . . . . . . . . . . . . . . . . . 141

10.3.1. The equicoercivity of the composite Gibbs-Landau free energy functionals . . . . . . . . 142

10.3.2. The Γ-limit of exchange energy functionals Eε . . . . . . . . . . . . . . . . . . . . . . . . . . 142

10.3.3. The continuous limit of magnetostatic self-energy functionals Wε . . . . . . . . . . . . . . 145

10.3.4. The continuous limit of the anisotropy energy functionals Aε . . . . . . . . . . . . . . . . 147

10.3.5. The continuous limit of the interaction energy functionals Zε . . . . . . . . . . . . . . . . 148

10.3.6. Proof of Theorem 10.1 completed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

11. Basic equations for Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . 149

11.1. The Landau-Lifshitz-Gilbert equation for magnetization dynamics. . . . . . . . . . . . . . . . . 149

11.1.1. The Landau-Lifshitz equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

11.1.2. The Landau-Lifshitz-Gilbert equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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11.2. Spatially uniform magnetization dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

11.2.1. Magnetization switching process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

11.3. Spin-momentum transfer in magnetic multilayers: Landau-Lifshitz-Slonczewski equation. . . 154

11.3.1. Landau-Lifshitz-Gilbert equation with Slonczewski spin-transfer torque term. . . . . . . 155

12.Current-driven microwave-assisted

Magnetization Switching. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

12.2. The analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

12.3. The numerical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

13. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

13.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

13.2. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

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6

1Introduction

1.1 Physical motivations: Weiss-Heisenberg theory.

A ferromagnetic material may be defined as one that possesses a spontaneous magnetization: that is,

sufficiently small volumes of it have an intensity of magnetization (magnetic moment per unit volume)

|M| :=Ms(T ) dependent on the temperature but independent, or at least only slightly dependent on the

presence or absence of an applied magnetic field1.1 [Bro62b, Bro63].

Modeling of ferromagnetic materials is not as natural as it sounds and has experienced over the years

many variations and changes. The modern understanding of magnetic phenomena in condensed matter

originates from the work of two Frenchmen: Pierre Curie (1859-1906) and Pierre Weiss (1865-

1940). Curie examined the effect of temperature on magnetic materials and observed that magnetism

disappeared suddenly above a certain critical temperature (nowadays known as Curie temperature) in

materials like iron. Weiss, in an effort to justify the existence of a spontaneous magnetization, proposed a

theory of magnetism based on the molecular field postulate, i.e. on the presence of an internal molec-

ular field proportional to the average magnetization that spontaneously align the electronic micromagnets

in magnetic matter. The theoretical investigations of Werner Heisenberg (1901-1976), replaced the

mysterious Weiss molecular field with the quantum mechanical effect known as exchange interaction,

which is less mysterious or more according to one’s feeling toward quantum mechanics. But this theory ,

based on exchange forces that tend to align the spins and thermal agitation that tends to misalign

them, says nothing about the direction of the vector magnetization M; only that its magnitude must

be Ms(T ) [Bro62b, Bro63].

Experimentally, it is observed that though the magnitude of |M| =Ms(T ) is uniform throughout a

homogeneous specimen at uniform temperature T , the direction of M is in general not uniform, but varies

from one region to another, on a scale corresponding to visual observations with a microscope. Uniform

1.1. By applied magnetic field we shall always mean the field of magnetizing coils or magnets (or both) externalto the specimen, as distinguished from the field (be it the H field or the B field) produced by the magnetization of thespecimen under consideration.

7

of direction is attained only by applying a field, or by choosing as a specimen, a body which is itself of

microscopic dimensions (a fine particle); the evidence of uniformity in the latter case is indirect but

convincing (see Chapters 8 and 9). The tendency of a ferromagnetic specimen to break up into domains,

with their vector magnetization oriented differently in any such domain, explains the possibility of a

demagnetized state; and in fact such a domain structure was postulated by Weiss in order to reconcile his

theoretically predicted spontaneous magnetization with the experimental possibility of demagnetization.

Today the evidence of domain structure are so many and so inescapable that its status is no longer that

of a postulate, but rather of an experimental fact (see Figure 1.1).

200µm

100µm

200µm

Figure 1.1. Domains observed with magneto-optical methods on homogeneous magnetic samples. (a) Images fromtwo sides of an iron whisker, combined in a computer to simulate a perspective view. (b) Thin film NiFe element(thickness 130 nm) with a weak transverse anisotropy.

In two respects, however, the range of validity of this fact has at times been supposed more universal

than it actually is. First, domains were for a long time tacitly assumed to be present in all specimens,

regardless of their geometry. This naive assumption delayed the theoretical understanding and practical

application of the properties of fine particles. Second, domains have often been discussed as if they were a

phenomenon to be expected in all ferromagnetic materials. Actually, both theory and experiment indicate

that domains in the usual sense – regions within which the direction of the spontaneous magnetization

is uniform or at least nearly so – do not occur unless there are present strong «anisotropy» forces,

which cause certain special directions of magnetization to be preferred. When such forces are absent or

weak, the magnetization direction, over dimensions comparable with the usual domain dimensions, varies

gradually and smooth. It is therefore clear that domain structure, though normal, is not universal. The

general problem to be examined in Micromagnetic theory is the problem of developing a theory of this

magnetic microstructure, concerning which the Weiss-Heisenberg theory is noncommittal: domains, when

they exist, should in principle emerge from the theory without having to be postulated. But has Brown

points out in [Bro63]: «No claim is made that Micromagnetic theory has been fully developed; all that

can be said is that the foundations have been laid». In this respect, one of the main aim of the research

activity that will be illustrated in the following chapters, is to gain a step further in the development of

Introduction

8

Micromagnetic Theory.

1.2 Technological Motivations

The study of ferromagnetic materials and of their magnetization processes has been, in the last sixty

years, the focus of considerable research for its application to magnetic recording technology. Indeed,

ferromagnetic media, below the Curie temperature Tc, possess a spontaneous magnetization state

in the absence of an applied field – which is the result of «spontaneous» alignment of the elementary

magnetic moments that constitute the medium – that, roughly speaking, can be changed by means of

appropriate external magnetic fields. The magnetic recording technology exploits the magnetization of

ferromagnetic media to store information [d’A04].

Magnetic recording technology can be tracked back to the idea of audio recording on metallic wires by

Valdemar Poulsen who, in 1898, demonstrated the possibility of magnetic recording via his telegra-

phone device. Further development led to the invention (around 1930) of magnetic tapes, which consisted

of fine ferromagnetic particles embedded in a non-magnetic film. Since the introduction of computers,

magnetic tapes become to be used for digital data storage. However, data storage on tape is limited to

sequential access, which involves repeated fast forward and rewind actions. Thus, the introduction of the

first hard disk drive by IBM in 1956 led to a substantial gain in speed as it allows for random access to

the data: this hard disk drive, featured a total storage capacity of 5MB at a recording density of 2kbit/in2.

In the quest to lower the cost and improve the performance, the areal density, i.e the number of bits per

square inch (b/in2), has increased by a factor of more than 200 million from 2 kbit/in2 to 500 Gbit/in2.

Nonetheless, the pursuit of higher areal densities still continues, and system designs have been discussed

for Tbit/in2 densities [MTM+02].

This astonishing rate of increase in areal density has required continuous scaling of the critical compo-

nents and dimensions of the magnetic recording devices, with the result that modern recording technology

has to deal with magnetic media whose characteristic dimensions are in the order of microns and sub-

multiples. Therefore, the design of nowadays magnetic recording devices requires a deep investigation of

the microscopic phenomena occurring within magnetic media [d’A04, PC11].

Recently the possibility to realize magnetic random access memories (MRAMs), has given

an additional impetus to this research field. The MRAM chips have many advantages over their silicon

counterparts, especially that of requiring energy only to change the value of the bits and not to maintain

the storage. They do not require refresh – since the information is stored as the magnetization state of

a permanent magnet – and moreover, unlike conventional silicon RAM, they retain data after a power

supply is turned off. Finally, MRAM requires only slightly more power to write than read, and no change

in the voltage, eliminating the need for a charge pump. This leads to much faster operation, lower power

consumption, and an indefinitely long «lifetime», so that it is estimated that such component should

rapidly replace the traditional memory in the next few years [CF07].

1.2.1 The modern recording process: hard disks and MRAMs.

Both hard disks and MRAMs rely on flat pieces of magnetic materials having the shape of thin-films.

Typically, the information, coded as bit sequences, is connected to the magnetic orientation of these films,

which have dimensions in the order of microns and submultiples.


9

N N N N N N SSSSS

Track widthShield

Read element

MR or GMR

sensor

Read current Write current

Magnetization Inductive write

element

Recording

medium

N S S

Figure 1.2. Simple representation of Read/Write longitudinal magnetic recording device present in hard disksrealizations.

A simple representation of a hard disk is shown in Figure 1.2. The recording medium is a flat

magnetic material that is a thin-film shaped magnetic medium. The read head and write head are

separate in modern devices, since they use different mechanisms. Indeed the write head is constituted by

a couple of polar expansions made of soft materials, excited by the current flowing in the writing coil.

The fringing field generated by the polar expansions is capable to change the magnetization state of the

recording medium. Generally the recording medium is made with magnetic materials that have privileged

magnetization directions. This means that the recording medium tends to be naturally magnetized either

in one direction (let’s say «1» direction) or in the opposite («0» direction). In this sense, pieces of the

ferromagnetic material can behave like bistable elements. The bit-coded information can be therefore

stored by magnetizing pieces of the recording medium along directions 0 or 1 [d’A04, TB00]. The size of

the magnetized bit is a critical design parameter for hard disks. In addition, for the actual data rates,

magnetization dynamics cannot be neglected in the writing process.

The reading mechanism currently relies on a magnetic sensor, called spin valve, which exploits the

giant magneto-resistive effect (GMR), i.e. a quantum mechanical magneto-resistance effect observed

in thin-film structures for whose discovery (in 1988) Alber Fert and Peter Grünberg were awarded

the 2007 Nobel Prize in Physics. Basically, the spin valve is constituted by a multilayers structure. Typi-

cally two layers are made with ferromagnetic material. One is called free layer since its magnetization can

change freely. The other layer, called pinned layer, has fixed magnetization. If suitable electric current

passes through the multilayers, significant changes in the measured electric resistance can be observed

depending on the mutual orientation of the magnetization in the free and pinned layer. Let us see how

this can be applied to read data magnetically stored on the recording medium. Basically, the spin valve is

placed in the read head almost in contact with the recording medium. Then, when the head moves over

the recording medium, the magnetization orientation in the free layer is influenced by the magnetic field

produced by magnetized bits on the recording medium. More specifically, when magnetization in the free

layer and magnetization in the pinned layer are parallel, the electrical resistance has the lowest value.

Conversely, the anti-parallel configuration of magnetization in the free layer and pinned layer yields the

highest value of the resistance. Thus, by observing the variation in time of the electrical resistance (that

Introduction

10

is, the variation of the read current passing trough the multilayers) of the GMR head, the bit sequence

stored on the recording medium can be recognized [d’A04, TB00].

1

0

Bit lines

Cross point

architecture

Word lines

Figure 1.3. Principle of MRAM, in the basic cross-point architecture. The binary information 0 and 1 is recordedon the two opposite orientations of the magnetization of the free layer of magnetic tunnel junctions (MTJ), whichare connected to the crossing points of two perpendicular arrays of parallel conducting lines. For writing, currentpulses are sent through one line of each array, and only at the crossing point of these lines is the resulting magneticfield high enough to orient the magnetization of the free layer. For reading, the resistance between the two linesconnecting the addressed cell is measured.

In a typical MRAM device, the binary information 0 and 1 is stored in elementary cells that can be

addressed via two perpendicular arrays of parallel conducting lines (bit lines and word lines in Figure

1.3). The reading mechanism is based on GMR effect, i.e the resistance between the two lines connecting

the addressed cell is measured. For writing, an MRAM cell can be switched by means of the magnetic field

pulse produced by the sum of horizontal and vertical current. This magnetic field pulse can be thought

as applied in the film plane at 45 off the direction of the magnetization. In this situation, the magnetic

torque, whose strength depends on the angle between field and magnetization, permits the switching of

the cell [CF07].

This behavior is simple in principle, but it is very hard to realize in practice on nanoscales. In fact,

the array structure must be designed such that the magnetic field produced by only one current line

cannot switch the cells. Conversely, the field produced by two currents must be such that it switches

only the target cell. Recently, to circumvent the problems of switching MRAMs cells with magnetic

field, the possibility of using spin-polarized currents, injected directly in the magnetic free layer

with the purpose to switch its magnetization, has been investigated. In particular, this possibility has

been first predicted by the theory developed by J. Slonczewski in 1996 [Slo96] and then observed

experimentally [Sun99]. The interaction between spin-polarized currents and the magnetization of the free

layer is permitted by suitable quantum effects. From a «macroscopic» point of view, these effects produce

a torque acting on the magnetization of the free layer. The resulting dynamics may indeed exhibit very

complicated behaviors.

The above situations are only few examples of technological problems which require to be investigated

by means of theoretical models. Now, referring to hard disk technology, at the present time the main

challenges and issues can be summarized as follows:

• Higher areal density.


11

• Improved thermal stability of magnetized bits.

• Increasing read/write speed in recording devices.

The first two points are strongly connected, since the smaller is the size of the bit, the stronger are the

thermal fluctuations which tend to destabilize the configuration of the «magnetized bit». The future per-

spectives in hard disk design show that the use of perpendicular media, patterned media and heat-assisted

magnetic recording technology will possibly yield areal densities towards 1Tbit/in2. Thus, being the

spatial scale of magnetic media in the order of, more or less, hundred nanometers, magnetic phenomena

has to be analyzed by theoretical models with appropriate resolution. This is the case of Micromagnetics,

which is a continuum theory that stands between quantum theory and macroscopic theories like mathe-

matical hysteresis models (Preisach, etc.). Moreover, as far as the read/write speed increases (frequencies

in the order of GHz and more), dynamic effects cannot be neglected. Therefore, as a result, the design

of modern ultra-fast magnetic recording devices cannot be done out of the framework of a rigorous

mathematical theory of ferromagnetic materials on mesoscopic scale. This is the technological motivation

for the research activity that will be illustrated in the following chapters.

1.3 Overview of the thesis

Chapter 2 is devoted to a brief review of some fundamental magnetostatic concepts. Chapter 3 is

dedicated to a rigorous introduction to potential theory, which are fundamental in deriving the main

properties of the demagnetizing field. In Chapter 4, the main properties of the demagnetizing field are

investigated, while Chapter 5 is dedicated to the study of the interesting properties of the demagnetizing

field when the magnetization is defined in a general ellipsoidal domain. This leads to the introduction

of the so-called demagnetizing factors which play a fundamental role in Micromagnetics. Chapter 6 is

devoted to the introduction of Micromagnetic Theory. The approach is variational in nature and based on

classical thermodynamic considerations. In Chapter 7 we start dealing with global and local minimizers

of the Gibbs-Landau free energy functional. More precisely, first and second order (external) variations

are introduced, and their role in the analysis of local equilibrium states is highlighted in the context of

ellipsoidal geometries. Chapter 8 is dedicated to the study of global minimizers of the Gibbs-Landau

free energy functional. More precisely, this chapter is devoted to the proof of generalization of Brown’s

fundamental theorem of the theory of fine ferromagnetic particles to the case of a general ellipsoid [Fra11].

Chapter 9 focus on the study of local minimizers of the Gibbs-Landau free energy functional. The

purpose of Chapter 10 is to rigorously derive the homogenized functional of a periodic mixture of

ferromagnetic materials. We thus describe the Γ-limit of theGibbs-Landau free energy functional, as the

period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to 0. Chapter

11 is devoted to the introduction of the Landau and Lifshitz model for the magnetization dynamics.

Eventually Chapter12 concerns the presentations of the results exposed in [dDS+11] where the current-

driven microwave-assisted switching process of a uniformly magnetized spin-valve is investigated.

Introduction

12

2Basic Magnetostatic Concepts

«I am not going to be able to give you an answer to why magnets attract each other except to tell you that they do. Andto tell you that’s one of the elements in the world. I could tell you that magnetic forces are related to the electrical forces

very intimately, but I really can’t do a good job, any job, of explaining magnetic forces in terms of something else you’remore familiar with, because I don’t understand it in terms of anything else that you’re familiar with.»

Richard P. Feynman

2.1 The Lorentz force and the magnetic induction field B.

We begin with a review of the theory of magnetic forces between conduction currents flowing in filamen-

tary circuits and permanent magnets. To this end let us consider the standard (oriented) affine space R3

in which a reference frame has been chosen (the laboratory frame). Let us suppose that some region

of the space Ω is occupied by a certain number of electrical circuits and permanent magnets. We perform

the following experiment: given a small filamentary test circuit c in which the stationary current I flows,

we consider in it a small rectilinear part, that we represent as a vector ℓ, which is connected to the rest

of the circuit c by flexible connections. By the means of a dynamometer we can measure the force F that

affects c due to the presence of the electrical circuits and permanent magnets in Ω. Assuming that ℓ is

electrically neutral we find that the element of wire ℓ, when placed in proximity of the current carrying

circuits, is subjected to a position dependent force F having the following characteristics:

a. At every point x ∈ R3\Ω the force F (x)(ℓ) depends on the orientation of ℓ but it is linear with

respect to ℓ. In other terms for every x∈E the map ℓ∈R3 7→F (x)(ℓ)∈R

3 is a linear operator.

b. The direction of F (x)(ℓ) is orthogonal to ℓ: in other terms

F (x)(ℓ) · ℓ= 0 ∀ℓ∈R3. (2.1)

c. The modulus |F (x)(ℓ)| is proportional to I |ℓ| where I is the intensity of the stationary current

flowing in the test wire c.

The phenomenology described in a. and b. can be mathematically characterized by saying that for every

x∈R3\Ω the force F is a linear antisymmetric operator on R

3. Indeed it is simple to show that the

orthogonality condition (2.1) is mathematical equivalent to the antisymmetric condition

F (x)(ℓ1) · ℓ2 =−ℓ1 ·F (x)(ℓ2) ∀ℓ1, ℓ2∈R3. (2.2)

13

Therefore due to the previous proposition and the phenomenologically observation c. we get that for every

x∈R3\Ω the force exerted on the elementary portion ℓ of the filamentary test circuit c can be expressed,

in an orthonormal basis e1, e2, e3, by the means of an antisymmetric matrix B(x) in the form

F (x)(ℓ) = IB(x)ℓ with B(x)=

0 −B1(x) −B2(x)B1(x) 0 −B3(x)B2(x) B3(x) 0

. (2.3)

The entries of the B(x) matrix can be computed by the use of filamentary circuits directed along the

orthonormal basis vectors. Indeed we have

B1(x)=B(x)e1 · e2 , B2(x)=B(x)e1 · e3 , B3(x)=B(x)e2 · e3. (2.4)

The unique antisymmetric second order tensor B(x) defined by (?) is called the magnetic flux density

field or magnetic induction field. Once oriented the affine space R3, there exists a unique vector field

B(x) such that

F (x)(ℓ)= Iℓ×B(x) (2.5)

and in an orthonormal basis the components of B(x) are expressed by the three characteristic entries

of the matrix representation B given by (2.3). Equation (2.5) is known as Laplace’s second law and

constitutes the operational definition of the magnetic induction field B at a point x∈R3\Ω.

q

B

v

F = qv×B

c1

c2

Figure 2.1. According to Lorentz’s force law (2.7) a charge which does not move, is not subject to any magneticinduced force while, when it moves, it is subjected to a force which is perpendicular to its velocity.

If we denote by N the number of free charge carriers flowing in the elementary portion ℓ of the test

circuit c, and indicated with q and vd(ℓ) the charge and the average velocity (drift velocity) of any such

free charge carriers, then from the Laplace’s second law (2.5) we get

F (x)(ℓ) =Nqvd(ℓ)×B(x). (2.6)

According to this relation, we expect that a single free charge q, moving with velocity v(x) in the presence

of the magnetic induction field B(x), is subject to a force F (x)(v) given by

F (x)(v)= qv(x)×B(x). (2.7)

Basic Magnetostatic Concepts

14

The previous relation, which gives the force exerted on a moving charge q by the magnetic induction field

B is called the Lorentz’s force law. We note that according to Lorentz’s force law (2.7) a charge which

does not move, is not subject to any magnetic induced force while, when it moves, it is subjected to a

force which is perpendicular to its velocity. The physical dimensions of the magnetic induction field are

[B] =

[

force

charge · velocity

]

=

[

m

Qt

]

. (2.8)

The unit of the magnetic induction field, in the modern metric system S.I., is the derived unit called

Tesla, defined by the position

N

C· s

m=

Volt

m· s

m=

Weber

m2=Tesla=T, (2.9)

where we have defined the Weber as the product Volt per second (s).

Remark 2.1. The Lorentz’s force law permits an operational definition of the magnetic induction

field based on the use of a test charge rather than a test circuit. To measure B in this way, although

experimentally more difficult, it is conceptually more correct. Indeed, while Laplace’s second law is valid

only when the B field does not undergoes appreciable variation along the elementary part ℓ of the test

circuit c, the expression of the Lorentz force is (physically) local.

2.2 The fundamental equations of magnetostatics in free space.

In the previous section we have seen how the phenomenology of magnetostatic interactions has lead to

the operational definition of a new physical quantity, known as magnetic induction field; but at this stage

we still do not know how to predict the B field generated in all space by a permanent magnet or by a

stationary circuit.

2.2.1 Laplace’s first formula.

Historically, the search for an analytic expression of the B field generated by circuits and permanent

magnets was one of the most investigated problems of the nineteenth century physics. In particular, the

study of the magnetic fields generated by circuits of many different shapes was one of the main research

topics of greatest physicists as Ampere, Arago, Biot, Savart and many others. The set of all their

experimental findings is synthesized in the so called Laplace’s first formula (sometime also called Biot-

Savart law):

B(x)=µ0

4π

∫

Ω

J(y)×∇y

(

1

|x− y |

)

dy ∀x∈R3\∂Ω, (2.10)

where Ω is the region of space in which the current density J is flowing, and ∂Ω the boundary of Ω.

Although the Biot-Savart law (2.10) is mathematically meaningful for every x∈R3\∂Ω, any extrapolation

of (2.10) to points inside the magnetized body constitutes an arbitrary definition, since the experimental

basis of the formulas does not include any information about force exerted by the body on one of its own

parts, or vice versa. Such extrapolations, nevertheless, are useful for mathematical completeness; whether

any physical significance is to be attached to the resulting expressions is a question to be investigated

later.


15

The physical constant µ0, commonly called the permeability of free space or magnetic constant,

is an ideal physical constant, which is the value of magnetic permeability in a classical vacuum. In S.I.

units, µ0 has the exact defined value:

µ0 := 4π · 10−7 W · sm

= 4π · 10−7Henry

m= 4π · 10−7 N

A2, (2.11)

where we have denote by «Henry» the product «Ohm» per «second».

2.2.2 Gauss’s law for magnetism

Under suitable regularity assumptions concerning the density current J and the domain Ω in which it

flows, it is simple to show that equation (2.10) can be recasted in the form

B(x)= curlA(x) with A(x) :=µ0

4π

∫

τ

J(y)

|x− y | dy . (2.12)

The vector A arising in the previous expression, is known as the magnetic vector potential. The

magnetic induction field B, being a curl field, it is necessarily divergence free. We are so lead to the well

known Gauss’s law for magnetostatics

divB= 0 . (2.13)

This equation, whenever considered from a classical mathematical point of view (i.e. in not some weak

sense) must be considered satisfied in all points of the space not lying on the boundary of the region occu-

pied by the stationary currents; indeed at these points (the ones on the boundary) a jump discontinuity

of the magnetic induction field B may arise.

2.2.3 Ampère’s circuital law

To go a step further in the mathematical and physical properties of the magnetic induction field we

introduce in an informally way a classical result of potential theory which will be stated and proved

rigorously in the next chapter: the reconstruction of a vector field in R3 from the knowledge of its curl and

divergence. Precisely, given a scalar field ρ and a vector field j, we want to find a vector field b such that

divb= ρ

curl b= jin R

3. (2.14)

It can be shown that under suitable regularity conditions concerning the differentiability and the order of

decay at infinity of the scalar field ρ and of the vector curl j, there exists a unique solution vanishing

at infinity of the system (2.14) and satisfying the continuity condition

divj= 0 in R3. (2.15)

This unique solution can be expressed in integral form as:

b(x) :=1

4π

∫

R3

curl j(y)

|x− y | dy−∇ 1

4π

∫

R3

ρ(y)

|x− y | dy. (2.16)


16

An applications of this result to our context, certainly possible since for stationary currents J the

continuity condition divJ=0 is satisfied, immediately leads to the well known Maxwell’s fourth law,

also known as Ampère’s circuital law

curlB= µ0J . (2.17)

As before, the previous equation, whenever considered from a classical mathematical point of view (i.e.

in not some weak sense) must be considered satisfied in all points of the space not lying on the boundary

of the region occupied by the stationary currents; indeed at these points (the ones on the boundary) a

jump discontinuity of the magnetic induction field B may arise.

The system of equations (2.13) and (2.17), together with the continuity equation divJ=0 and suitable

transmission conditions on the boundary of the region occupied by the body, constitute the fundamental

equations of magnetostatics in free space.

2.2.4 Ampére equivalence theorem

We now consider the magnetic vector potential and the magnetic induction generated by a physically

small circuit (that is, one whose linear dimensions are small in comparisons with its distance from the

other circuits with which interacts) at a point distant from the circuit. We know from (2.12) that the

magnetic vector potential A0 due to such a small circuit occupying the region Ω is given by

A0(x) =µ0

4π

∫

Ω

J(y)

|x− y | dy. (2.18)

If we suppose Ω to be a toroidal region, of major and minor radius R and r, centered around the origin

o of the reference frame, then a direct computation shows that the limit for R→ 0 computed with the

constraint that the aspect ratio remain constant, leads to

limR→0

A0(x) =µ0

4π

m× r|r |3 (2.19)

where r = (x − o) is the position vector of the point x ∈ R3 and m is the so called magnetic dipole

moment associated to Ω given by

m=1

2

∫

Ω

(y− o)×J(y) dy. (2.20)

By taking the curl of A0 we so finish with

B0(x)=µ0

4π

(

m

|r |3 −3(m · r)r

|r |5)

, r := (x− o). (2.21)

The result thus found is a special case of the Ampère equivalence theorem to state which we must

recall the early stages of magnetic theory: once introduced magnetic poles p1, p2 as fictitious entities

having the same modulus p but different sign, and such that if they occupy the positions x and x + h

exert mutual forces according to Coulomb’s law:

F 21=µ0

4π

p1p2

|h|3h , (2.22)


17

the quantity m := ph was defined as the magnetic doublet. Then the ideal magnetic dipole of

moment m was defined as the limiting case of a pair of magnetic poles (of different sign) and of strength

|m| whose distance shrinks to zero, but in such a way that the modulus |m| of m remains constant. The

Ampère’s equivalence theorem states that: at points far from a small current loop, current loops behaves

like a permanent magnet of moment m. Moreover, the magnetic vector potential A originated from a

smooth distribution of dipoles m in Ω is given, for every point x not belonging to the boundary of Ω by

Am(x)=µ0

4π

∫

Ω

curlm(y)

|x− y | dy+µ0

4π

∫

∂Ω

m(y)×n(y)

|x− y | dσy. (2.23)

Although was once customary to base the theory of material magnetism on the magnetic pole concept,

nowadays is more fashionable (as we did) to base the theory on Amperian currents – if a current I

flows in the positive direction around the contour of a vector area σ, the circuit is said to have a magnetic

momentm := Iσ. Indeed, since ordinary conduction currents an their magnetic fields must be considered

along with magnetic matter, the Amperian current method provides a more unified theory. That’s why we

have based our considerations on the Amperian current concept. Nevertheless, in ferromagnetism, poles

(as we will see) are more useful than Amperian currents and the reason for this is that the constitutive

relations linking macroscopic and microscopic fields can be better derived reasoning in terms of moments.

That is why we shall regard magnetic moment itself as the physically fundamental concept in material

magnetism, and we shall also derive and use the mathematically equivalent pole formulas. By the way, it

is important to stress that such a choice has nothing to do with any «real» or «fundamental» character

of either poles or Amperian currents.

2.2.5 A first look to the demagnetizing factors

Because of its relation to certain formulas for magnetic specimens, we note here the formula for the

internal flux density of a surface-current distribution on the surface of an ellipsoid. Since in a later chapter

we will investigate the problem in rigorous mathematical terms, let us now focus the problem in more

physical terms.

e3

e1

e2

III

Figure 2.2. It turns out that if a constant (filamentary) current I flow clockwise about the e3 direction along anyperpendicular slice (with respect to e3) of the ellipsoidal surface, then the flux density inside the ellipsoid is uniformand directed along the e3 direction.


18

To this end let us consider a triaxial ellipsoid of principal axes directions e1, e2,e3. It turns out that

if a constant (filamentary) current I flow clockwise about the e3 direction along any perpendicular slice

(with respect to e3) of the ellipsoidal surface, then the flux density inside the ellipsoid is uniform, directed

along the e3 direction, and equal to

B= µ0I(1−N3)e3, (2.24)

where N3 is a geometrical factor determined by the axis ratios of the ellipsoid. Factors N1 and N2

corresponding to the other two axes may be similarly defined. The three ’s satisfy

N1 +N2 +N3 = 1, (2.25)

and as any ellipsoid axis becomes infinite, the corresponding demagnetizing factor approaches zero. It

follows from these statements that for a sphere, each demagnetizing factor is1

3; and that for an infinite

circular cylinder with axis along e3, N1 =N2 =1

2and N3 = 0. We shall encounter these same factors in

connection with ellipsoidal magnetic specimens, under the name demagnetizing factors.

2.3 Magnetized Matter

A realistic theoretical interpretation of magnetic phenomena in matter must be based at essential points

on atomic concepts. Ideally these would be treated by rigorous quantum mechanical methods. In practice

this is not possible: quantum mechanics provides some of the fundamental concepts such as electron spin;

but for the practical treatment of a crystal containing many atoms it is not necessary, and for many

purposes sufficient, to use a classical approximation. In this respect, in what follows, we will base the

interpretation of magnetic phenomena in matter on the already mentioned Ampère’s equivalence theorem

(see subsection 2.2.4) according to which, at long distance, a coil traversed by a current behaves as a

magnetic dipole.

Indeed the electrons, which in the Rutherford-Bohr planetary model of the atom are in orbit

around the positively charged nucleus, are similar to microscopic coils traversed by currents (microscopic

currents), and therefore each of these electrons is equivalent to a magnetic dipole (due to Ampère

equivalence theorem). In the absence of a local magnetic field inside the material, all these microscopic

dipoles are randomly oriented: their resultant, performed on any little piece of matter, is therefore zero,

and the material does not generate any macroscopic magnetic effect. However, in the presence of a local

magnetic field in the matter, polarization phenomena arise: first of all for orientation polarization, but, as

we shall see, also due to different phenomena. Therefore the resultant magnetic moment of each portion

of material is no longer zero, and this causes both an alteration of the external magnetic field and a

mechanical action on the material by the same external field.

2.3.1 The fundamental equations of magnetostatics in matter

We have seen in the previous section that the fundamental equation of magnetostatics in free space are

given by:

divB= 0curlB= µ0J

, (2.26)

the vector J denoting the density of macroscopic currents, assumed known. Formally, in the investigation

of magnetic phenomena, the presence of matter, can be taken into account by a very simple change in

the system (2.26): indeed, everything still goes as if it were still free space, but with the presence now

2.3 Magnetized Matter

19

both of macroscopic conduction currents of density J and of many microscopic currents of atomic nature

of current density Jm. Therefore, in the presence of matter, the equations of magnetostatics become:

divB=0curlB= µ0(J+Jm)

. (2.27)

The difficulty now lies in the fact that while the density J associated to macroscopic currents is known,

the same can not be said for the density Jm of the microscopic currents, so that some additional effort

must be taken to let the system (2.27) to be of practical use.

The standard strategy in overcoming this kind of problems is to find a relationship that links the

microscopic density currents Jm to a measurable macroscopic quantity (therefore directly or indirectly

note). To this end we will proceed as follows: in the next section we introduce themagnetic polarization

vector (or magnetization) M and then determine a functional relation between M and Jm. Using

this relation, the system of equations (2.27) is transformed into a system of differential equations that

express B as a function of the only macroscopic quantities J and M. Therefore, if the magnetization

M (in addition to the macroscopic current J) is explicitly known, these equations allow, given suitable

transmission conditions and regularity conditions at infinity, to determine the magnetic induction field B.

The problem is that in most of the cases of interest the magnetization M is an unknown of the problem,

and it deeply depend on the magnetic induction field B in which the material is immersed, so that to

solve the system (2.27) a relation (constitutive relation) G must be found between the magnetization

M and the magnetic induction field B. Indeed, outside mathematical difficulties, the knowledge of the

constitutive equation M= G(B) permits to solve the system (2.27).

It is the search for constitutive relations the historically starting point of the road which brings to

Micromagnetic Theory; that is why we take the opportunity here to mix some history and physics.

2.3.2 Researching a constitutive relation: Lorentz and Weiss ideas, Micromagnetics.

The first interesting result in the search for a constitutive relation dates back to Lorentz who introduced

the concept of magnetic local field BL, as a semi-classical bridge between the microscopic world of

microscopic currents densities Jm and the macroscopic world of measurable macroscopic quantities M

and B. In more formal terms, the idea of Lorentz can be summarized as the searching for the following

system of functional dependencies: M=F(BL,B) and BL=L(M), so that the solution of the previous

system of two equation gives the wanted constitutive relation. As we will see in the next section, the

functional dependencies introduced by Lorentz, although classical and linear in nature, can take into

account many aspects of the phenomenology of paramagnetic and diamagnetic materials.

More effort must instead be put into understanding which are the constitutive relations governing

ferromagnetic phenomena, in which a spontaneous alignment of atomic moments can arise at room

temperature. Historically speaking, is due to Pierre-Ernest Weiss the attempt to extend Lorentz

results to the explanation of the behavior of ferromagnetic materials. In 1906, he suggested the existence

of magnetic domains in ferromagnets – i.e. the view of a ferromagnetic material as a partitioned

structure in which blocks (domains) are made of small regions in which the magnetization is uniform –

and, in the attempt to explain the reason for such a spontaneous alignment of atomic moments within

a ferromagnetic material, came up with a semi-phenomenological theory nowadays known as Weiss

molecular field theory: he considered a given magnetic moment in a material experienced a very high

effective magnetic field (the Weiss molecular field) due to the magnetization of its neighboring spin, and

assumed that the intensity of the intensity of the molecular field is proportional to the magnetization.


20

In spite of its great success in explaining some ferromagnetic phenomena on mesoscopic scale, Weiss

theory is silent on the physical origin of the molecular filed, and only twenty years later, in 1928, Heisen-

berg showed that the strong tendency that atomic magnetic dipole moments have to align into a common

direction is due to an entirely non-classical phenomenon which he called exchange interaction. We

will attempt later to explain the nature of this interaction. From the time being we want only point

out that although the existence of spontaneous magnetization in ferromagnetism is explained by the

Heisenberg-Weiss molecular field postulate, this theory does not let predict anything about the direction

of the magnetization vector M: indeed it only explain why its magnitude must be constant at a given

temperature.

Now, in general, the direction of M is not uniform – it varies on scales corresponding to visual

observation with a microscope – and the distribution of the direction of the magnetization inside a

ferromagnetic body is a kind of information which is essential in practical applications. It is in this spirit

that William Fuller Brown developed a theory of fine ferromagnetic particles in which the possible

magnetization states, and therefore the possible magnetization configurations M, can be determined

by seeking for a minimum of a suitable energy functional (Gibbs-Landau free energy functional)

which is the main topic of this thesis. The original idea of Brown was to set up a complete and rigorous

theory of all magnetization processes in any ferromagnetic materials, able to explain magnetic domains

and domain walls as a result of the micromagnetic theory, rather then to leave these concept as result of

pure physical intuition.

2.4 Classical aspects of atomic magnetism.

Before starting the treatment of the magnetism in matter under stationary conditions, according to the

what outlined in the previous paragraph, we introduce some concepts relating the behavior, in external

magnetic field, of an atom. In what follows we refer to the semi-classical Rutherford-Bohr planetary

model, according to which the atom consists of a nucleus with a massive positive charge Ze+ around

which, attracted by the Coulomb force, orbit (in steady state) Z electron describing elliptical orbits.

+

Ia e−

v0

Figure 2.3. According to the semi-classical Rutherford-Bohr planetary model, the atom consists of a nucleus witha massive positive charge Ze+ around which, attracted by the Coulomb force, orbit (in steady state) Z electrondescribing elliptical orbits.

It is well known that systems of geometric dimensions as small as atomic systems, in which the charac-

teristic dimensions are of the order of 10−10 meters, can not be treated with classical mechanics, but it is

necessary to use quantum mechanics. Therefore the Rutherford-Bohr model is to be considered a drastic

approximation: nevertheless it is able to account for several important aspects of the phenomenology.


21

2.4.1 The angular momentum µL

Let us consider, for simplicity, a hydrogen atom in its ground state and assume that the electron orbit is

circular. Let us also indicate with r0 the radius of the (circular) orbit, with me the electron mass, with

the e the modulus of the electron charge, with ω0 the angular velocity and with T0 the revolution period.

Taking into account that due to Coulomb law |F e|= 1

4πε0

e2

r02 , while due to Newton second law of motion

|F e|=me|a|=meω02r0, we get the relation

|F e|= 1

4πε0

e2

r02 =meω0

2r0≡ 4π2

T02 mer0 (2.28)

and therefore

T0 =4π

eπε0mer0

3√

. (2.29)

The radius of the orbit can be computed experimentally from a measurement of the ionization work of

the hydrogen atom, that is, from the energy which must be supplied to the electron to tear it from the

orbit and bring it to infinity. Indeed since a non moving electron has at infinity zero energy, the ionization

work Li must be equal in magnitude to the total energy ET that the electron has when it is linked to the

atom, so that Li+ET = 0. We now observe that due to (2.28) we get ET =− 1

8πε0

e2

r0and therefore

r0 =e2

8πε0

1

Li. (2.30)

Experimentally the ionization work of the hydrogen atom turns out to be Li⋍1, 35 eV. Therefore r0 ⋍0,

5 ·10−10m and T0⋍1,5 ·10−16 sec. But the fact that an electron orbits with period T0 around the nucleus,

implies that an e− charge passes trough every fixed point of the orbit T0−1 times per second, and this is

equivalent to an atomic current of modulus |Ia|= e

T0.

In the case of the hydrogen atom we so get an atomic current |Ia| ≃ 1mA and a magnetic moment

|m|= |Ia|S=e

T0πr0

2≃ 9, 35 · 10−24Am2, (2.31)

where we have denoted by S the area of the disk having the circular orbital motion of the electron as

boundary. The previous value, although derived from classical considerations, is in well agreement with

experimental results.

v0

Ia

+

e−

m

µL

Figure 2.4. he magnetic moment m due to the orbital motion of the electron (orbital current) is proportional toits angular momentum L with respect to the nucleus, but they are anti-parallel.


22

It is interesting to observe that the magnetic moment m due to the orbital motion of the electron

(orbital current) is proportional to its angular momentum L with respect to the nucleus: indeed we have

µL= r0×mev0 (2.32)

and therefore both µL and m are orthogonal to the orbit. But they are anti-parallel: indeed due to the

negative charge e− of the electron the orbital current is in the opposite direction with respect to the

electron velocity v0. Concerning their modulus we get |µL|= 2πmer02

T0and therefore, from (2.31)

m=e−

2meµL. (2.33)

From the previous relation we see that the ratio between the magnetic moment m and the angular

momentum µL is a function of the intrinsic electron properties only. In general, for every atomic system,

the ratio between m and µL is called gyromagnetic ratio and is denoted by g. In other terms the

gyromagnetic ratio is the proportionality constant such that m= gµL.

The gyromagnetic factor of the electron of hydrogen atom (in its ground state) is equal to

g=e−

2me. (2.34)

It is quite surprising that this conclusion, although derived from pure classical arguments, is still true

in the context of quantum mechanics, and is therefore applicable to the electron orbital moment of any

atomic system. On the other hand, according to quantum mechanics the orbital angular momentum (in

any atomic system) is constrained to assume values which are integer multiples of an universal constant

~, and therefore the modulus of the angular momentum µL must necessary have an expression of the form

|µL|= l~ = l

(

h

2π

)

, l∈N. (2.35)

The constant h = 6, 62617 · 10−34 Joule, appearing in the previous equation, is the well-known Plank

constant, while the non negative integer l is the so called orbital quantum number. Taking into

account equation (2.33) we so reach the conclusion that the modulus of the magnetic moment must

necessary be a non negative integer multiple of the quantity mB :=e−

2me~, which is known as Bohr

magneton:

|m|= lmB= le−

2me~ , l ∈N. (2.36)

2.4.2 The spin momentum

The total magnetic moment of an atomic system is not only due to the orbital momentum of electrons

in their revolution motion around the nucleus: atomic constituents are in fact also equipped with an

intrinsic moment (both of an intrinsic magnetic moment and of an intrinsic angular momentum) as if it

were small spheres, having a spatial distribution of charge and mass, rotating around a barycentric axis

(see Figure 2.5).


23

µS

Figure 2.5.

To the intrinsic moment (both intrinsic angular momentum and magnetic dipole

moment) is given the name of spin momentum. It is an experimental fact that

the spin angular momentum µS is the same for electron, proton and neutron, and

equal to |µS | = ~/2. On the other hand, although the spin angular momentum is

the same for all of these three particles, their intrinsic magnetic moment is not the

same, because they come out with different gyromagnetic factors. Experimentally

these turn out to be respectively:

ge := 2

(

e−

2me

)

, gp := 2, 79

(

e+

2mp

)

, gn := 1, 91

(

e−

2mp

)

(2.37)

where we have denoted by me the electron mass and by mp the proton mass. Therefore, taking into

account the expression |µS |=~/2 for the spin angular momentum, we finish with the following expressions

for the corresponding intrinsic magnetic moments:

|µe|= e−

2me~ , |µp|=

(

2, 79

2

)

e+

2mp~ , |µn|=

(

1, 91

2

)

e−

2mp~. (2.38)

We note that the intrinsic magnetic moment of the electron is equal to one Bohr magneton, id est it is

equal to the orbital magnetic moment of the electron of the hydrogen atom in its ground state. Since the

mass of the proton is almost two thousand times larger than that of the electron, the intrinsic magnetic

moment of nucleons (protons and neutrons) is about three orders of magnitude smaller than the one of

the electron, and its contribution can usually be neglected in most of the considerations concerning their

effect on the magnetized matter.

The total atomic magnetic moment of each atom is obtained as a vector sum of the magnetic

orbitals and spins. In doing this, however, some precise rules, established by quantum mechanics, must

be taken into account:

• Pauli exclusion principle. In an atomic system, no more than two electrons may occupy the

same quantum state simultaneously, and if it is the case then their spins must be anti-parallel.

• Quantized projection of the orbital angular momentum. The projection along an axis

(and in particular along the projection of a possible external magnetic field) of the orbital angular

momentum, can only take values which are: integer multiples of ~, and belong to the interval

[−l~, +l~]. On the other hand, the spin of electrons and nucleons can only be parallel or anti-

parallel to this direction.

In the computation (with these rules) of the orbital and spin magnetic moments, it turns out that many

atoms, characterized by a symmetrical spatial distribution, appear to have zero magnetic dipole moment,

and this conclusion is confirmed by experimental measurements. Even when the atomic magnetic moment

is different from zero, as a rule, in the absence of external magnetic field, the orientation of the moment

of the various atoms is completely random, so that each portion of matter has a result, zero macroscopic

magnetic moment.


24

To this rule make exception for the ferromagnetic materials, for which once induced, by an external

magnetic field, a preferred orientation of the elementary magnetic moments, this orientation remains to

some extent also by removing the external field, so that the material retains a non zero magnetic moment

even when an external magnetic field is no more present (permanent magnets). In any case, although with

strong differentiations for the various types of materials (diamagnetic, paramagnetic or ferromagnetic), an

external magnetic field has the effect of inducing a resultant magnetic moment which is non zero inside

the material.

2.5 The magnetization vector and its relation with microscopic currents.

The matter in the magnetic field can be thought of as a collection of atoms or molecules having non-

zero total magnetic moment. This amounts to thinking of existence, in the matter, of atomic microscopic

currents. Following the track outlined in section 2.3, we introduce the vector magnetization M, and

determine the relationships linking the macroscopic quantity M to the microscopic atomic currents Jm.

m1

m2

mi

Ω

Ωε(x)

x

Figure 2.6. The magnetization vector M(x) is the result of an average limiting process.

To this end we consider a region Ω occupied by a magnetic body. For every x∈Ω let us now consider

a filter base (Ωε(x))ε>0 of measurable sets converging to x; we then denote by Nε(x) the numbers of

microscopic magnetic dipoles contained in Nε(x) and withmi, i∈1,2, ...,Nε(x), their atomic magnetic

moments.

The magnetization vector M at the point x is then defined as

M(x) := limε→0

1

|Ωε|∑

i=1

Nε(x)

mi. (2.39)

and, in S.I. units, it is measured in amperes per meter (A/m). The existence of such a limit depends on

the choice of the family (Ωε(x))ε>0 and it must be considered an assumption of the theory the existence,

for every x∈Ω, of a family of neighborhood of x such that the previous limit exists.

For example, to let a uniform distribution of microscopic magnetic dipoles to give rise to an uniform

magnetization vector M, one must necessary search for a filter base converging to x and such that

limε→0Nε(x)|Ωε(x)|−1=1. In what follows we will therefore assume that such a filter base exists for every

x∈Ω, and moreover that the vector magnetization M so obtained, is a smooth function in Ω. Once the

existence of such a smooth function is postulated, the problem still remains on how to compute the limit

in (2.39). As we will see, it is one of the aim of the micromagnetic theory theory the determination of

the distribution of magnetization vector M, inside a magnetic body.

2.5 The magnetization vector and its relation with microscopic currents.

25

To see how the magnetization M is related to the microscopic atomic currents (Amperian currents)

we have to recall that the magnetic vector potential A originated from a distribution of dipoles M in Ω

is given, for every point x not belonging to the boundary of Ω by

AM(x)=µ0

4π

∫

Ω

curlM(y)

|x− y | dy+µ0

4π

∫

∂Ω

M(y)×n(y)

|x− y | dσy (2.40)

while the the vector potential AJ originated from a distribution of dipoles currents in Ω is given, for every

point x not belonging to the boundary of Ω by

AJ(x)=µ0

4π

∫

Ω

J(y)

|x− y | dy+µ0

4π

∫

∂Ω

K(y)

|x− y | dσy. (2.41)

Therefore, imposing the equality AJ = AM to be verified for every bounded domain Ω occupied by a

magnetic body, we finish with the following relations

J= curlM , K =M×n (2.42)

which constitute the desired linking between the magnetization vector M and the microscopic density

currents J and K. Due to the previous relations the fundamental equations of magnetostatics in matter

(2.26) become

divB= 0curlB= µ0J+ µ0 curlM

(2.43)

or equivalently

divB=0curlH=J

(2.44)

where we have denoted by H the magnetic field defined by the position

H=B− µ0M

µ0. (2.45)

We note that in free space the relation between B and H is of pure proportionality, the previous equation

reading in this case as B= µ0H.

Following the track already mentioned at the end of section 2.2.3, in order to make the system (2.43)

have a unique solution (when endowed with suitable interface conditions), we must find a constitutive

relation between the vectors B and M. To this end we will follow the idea of Lorentz, already outlined

in subsection 2.3.2, consisting in the searching for the functional dependencies: M = F(BL, B) and

BL = L(M), so that the solution of the previous system of two equation gives the wanted constitutive

relation.

2.6 The constitutive relation BL =L(M) between the local field and the magnetization.

For our present purposes of interpreting the mechanisms of magnetic polarization of matter, the problem

is to determine the stresses acting on each individual atom (or molecule). More precisely, we are interested

in determine the local field Hl generated, in the position occupied by that atom, by all other atoms as

well as from external sources.


26

As we shall see shortly, both the Lorentz and Weiss models are based on the assumption that the

local field Hl is expressible as a linear combination of the macroscopic vectors M and H, with coefficients

determined by phenomenological considerations. The procedure usually used for the calculation of the

relation linking Hl to H and M is described in the next subsection.

2.6.1 The Lorentz local field: the Lorentz sphere.

Consider a specimen Ω of arbitrary shape and assume that an external field H0 is being applied. The

macroscopic magnetic field located at a point x in Ω is the resultant of the externally applied field and

of the field due macroscopic distribution of magnetization M:

H(x)=H0(x)+Hd[M](x) (2.46)

where we have denoted by Hd[M] the macroscopic magnetic field produced by the distribution of magne-

tization in Ω. On the other hand, the local field located at a point x in Ω is the resultant of the externally

applied field and of the field due to all other dipole in the specimen. This «local field» intensity hi varies

rapidly with time, because of the thermal motion of the atoms; but presumably the contribution of all

except the very near atoms is subject to very small resultant fractional fluctuations, because of the large

number of atoms contributing and of the fact that the motions of any two of them, except two very close

together, are practically uncorrelated. The total local field intensity is, in a static approximation

Hl(x) =H0(x)+∑

i=/ x

hi(x) (2.47)

where hi(x) is the field intensity of dipole i at the position of dipole x.

To find a relation between the local field Hl at x and the macroscopic magnetic field H we will use

the Lorentz sphere method. We imagine to remove from Ω a small sphere ΩS centered on x with

a diameter much larger than the average distance between two atoms, but still small enough that the

magnetization in ΩS can be considered to be uniform in space. Such an intermediate distance exists

even for particles of linear dimensions as small as 1 micron (10−4cm), since the lattice spacing is of order

(10−8cm). However for particles of only 1/100 this size, Lorentz’s «physically small» sphere is only about

10 atoms across and extends 1/10 of the distance across this specimen; calculations based on it must then

be viewed with some skepticism.

ΩS

M

mx

Ω

Figure 2.7. The idea of used by Lorentz [1909] was to separate the dipoles i in (2.47) into two groups: thoseoutside the sphere ΩS about dipole x, and those inside that sphere. And then to evaluate the contribution of dipolesoutside the Lorentz sphere by replacing the sum by an integral.

2.6 The constitutive relation BL =L(M) between the local field and the magnetization.

27

Assuming that the magnetization in ΩS is not perturbed by the removal of ΩS, the macroscopic

magnetic field at x (due to the linearity of the magnetostatic integral operator Hd rigorously introduced

in Chapter 4) decomposed as (denoting by χΩSthe characteristic function of Ω):

H = H0 +Hd[χΩ\ΩSM] +Hd[χΩS

M]

= H0 +Hd[χΩ\ΩSM]− 1

3M

(2.48)

where the last equality is a consequence of the supposed uniformity of M inside ΩS.

On the other hand, it is well known how to evaluate the sum appearing in equation (2.47) when, for

a specified lattice structure and shape, all dipoles are equal and parallel. The idea of used by Lorentz

[1909] was to separate the dipoles i in (2.47) into two groups: those outside the sphere ΩS about dipole

x, and those inside that sphere. Granting that an ΩS meeting Lorentz’s specifications can be found, we

can evaluate the contribution of dipoles outside the Lorentz sphere by replacing the sum by an integral.

We so get

Hl(x) =H0(x)+Hd[χΩ\ΩSM](x) +

∑

i∈ΩS\x

hi(x). (2.49)

It can be shown that for a cubic lattice of equal and parallel dipoles, the sum in (2.49) vanishes. A sub-

stitution of (2.48) into (2.49) gives formula usually quoted, without sufficient statement of the conditions

for its validity, as the Lorentz local-field formula.

Hl=H +1

3M . (2.50)

In terms of the B vector it reads has

Hl= B− 2

3M. (2.51)

The two previous relations constitute the first of the two functional dependencies, M = F(BL,B) and

BL=L(M), outlined in section 2.2.3.2. The constitutive relation BL=L(M) will be achieved in the next

section.

Remark 2.2. If all the conditions just stated are satisfied except the condition that the crystal have

cubic symmetry, then the sum in (2.49) no longer vanishes, but the average of its three components still

vanishes. In this case the scalar factor1

3in (2.50) must be replaced by a tensor, having the symmetry of

the lattice, whose trace is1

3.

Remark 2.3. In ferromagnetic crystals the exchange forces hold the spins of neighboring atoms nearly

parallel; in consequence, the conditions for validity of Lorentz’s formula for cubic crystals, equation (2.51),

and its tensorial generalization for non-cubic crystals, may be expected to be reasonably well satisfied, at

least at low temperatures and in crystals in which there is a single lattice of dipoles. When the symmetry

is not cubic, the anisotropic term in the local-field formula may be omitted provided an equivalent term

is inserted in the crystalline anisotropy energy, to which there may be other contributions also.

2.6.2 The Weiss molecular field

As we will see in the next section, in paramagnetic and diamagnetic materials, the magnetization M (that

as we will is proportional to the local field Hl and therefore also the the macroscopic magnetic field H)

has a very small intensity compared to H. In other terms |M| ≪ |H| and therefore the 1/3 coefficient

appearing in the Lorentz local-field formula (2.50) thus not play a critical role, furthermore in many

applications it is possible to assume the equality between Hl and H.


28

Figure 2.8. To physically justify the so high values of γ, Weiss theorized that a ferromagnetic material can be seenas a partitioned structure in which blocks (Weiss domains) are made of small regions each of which is spontaneouslyuniformly magnetized with a magnetization intensity which is very near to its saturation value.

On the other hand, the analysis of experimental data concerning ferromagnetic materials show that

the 1/3 coefficient appearing in the Lorentz local-field formula (2.50) is no more in agreement with the

reality, and that it is necessary instead the use of a coefficient which is in the range [103,104]. The reason

of this strong magnetism was first clarified by P. Weiss in 1907. He assumed that in a ferromagnetic

materials there exists an effective field which he called molecular field Hw, which is still proportional

to the magnetization, and that must be superimposed to the Lorentz local-field in the description of

ferromagnetic phenomena. He so ended up with the microscopic constitutive relation

Hl=H+Hw with Hw := γM. (2.52)

The coefficient γ, known as Weiss constant, has a value (for ferromagnetic materials) which is much

larger than the Lorentz local-field coefficient 1/3 (actually γ typically range from 103 to 104), so that

the molecular field cannot be attributed to a classical magnetostatic interaction. To physically justify

the so high values of γ, Weiss theorized that a ferromagnetic material can be seen as a partitioned

structure in which blocks (Weiss domains) are made of small regions each of which is spontaneously

uniformly magnetized with a magnetization intensity which is very near to its saturation value (see

figure 2.8). These domains, for which the orientation is random in a non-magnetized material, can orient

themselves in the presence of a weak field also: indeed, in the presence of an external magnetic field

Ha, the domains magnetized parallel to Ha grow in volume, whereas those magnetized along a different

direction decrease in volume. Eventually, all domain can rotate and align with Ha. The existence of

Weiss domains with linear dimensions ranging from the fraction of µm and a few hundred µm, has been

detected experimentally; but their mechanism of formation is not interpretable in the framework of

classical physic and was clarified by Heisenberg, on the basis of quantum mechanics considerations, more

than twenty years after Weiss has advanced his hypothesis.

2.7 The constitutive relation M = F(BL, B) between the local-field Hl and the macroscopic

field M and B.

The last step in the track outlined at the end of section 2.2.3, in order to make the system (2.43) have

a unique solution (when endowed with suitable interface conditions), it remains to find the functional

dependence BL = L(M). To this end we now describe the two main microscopic contribution to the

magnetic properties of materials: the Larmor precession which constitutes the main contribution to the

macroscopic behavior of diamagnetic materials, and the Langevin function which, taking into account

orientation polarization phenomena, constitutes the main contribution in the macroscopic phenomenology

of diamagnetic and ferromagnetic materials.

2.7 The constitutive relation M=F(BL,B) between the local-field Hl and the macroscopic field M and B.

29

2.7.1 Larmor precession

As already seen such an electron is characterized by an angular momentum µL and by a magnetic moment

m0=−e

2meµL. Let us now suppose Bl= µ0Hl to be a local and uniform magnetic field acting on it. In the

hypothesis that the local field induces only a small perturbation in the motion, it is simple to deduce,

from classical considerations that

µL=m0×Bl=− e

2meµL×Bl , (2.53)

and therefore the motion of µL is a precessional one (Larmor precession) with angular velocity ωL=e

2meµL (Larmor angular velocity) and period TL= 2π/|ωL|.

Bl

m0

µL=m0×Bl

Figure 2.9. Larmor precession.

This precessional motion of the electron produces a current IL (Larmor

current) flowing counterclockwise around Bl (due to the negative charge

of the electron) and whose intensity is given by

|IL|= e

TL=

e2

4πmε|Bl|. (2.54)

It can be shown that under the assumption that the atom has an isotropic

space distribution, the magnetic momentmL associated to such a current

(known as Larmor magnetic moment), is given by

mL=−e2r2

6mεBl (2.55)

If now we denote by Z the atomic number and with ri the ray associated to the circular orbital motion

of the i-th electron (i∈1, 2, ..., Z), then the total Larmor magnetic moment reads as

mL=−Ze2a2

6mεBl , (2.56)

where we have denoted by a2 =1

Z

∑

i=1Z

ri2 the mean square value of the rays associated to the Z electron

orbits.

The Larmor precession is present in all materials, but in the paramagnetic and ferromagnetic mate-

rials, whose atoms have a non zero intrinsic magnetic moment, the magnetization by orientation is

dominant and this masks the diamagnetism due to Larmor precession.

2.7.2 Magnetization by orientation. Langevin function.

Since every atom has its own intrinsic magnetic momentm0, it is subject to orientation polarization due

to the presence of a local magnetic field Bl = µ0Hl. This kind of polarization can be described by the

means of statistical mechanics, but for our purposes it is sufficient to say that for isotropic materials at

temperature T , the effect of the local-field Bl := |Bl|el will be to average for an effective magnetic moment

〈m0〉, aligned along Bl, and given by

〈m0〉=L(y)(m0 · el)el. (2.57)


30

Here y :=m0 ·Bl

kBT, with kB the Boltzmann constant, and L(y) the Langevin function defined as

L(y)= coth y− y−1 (2.58)

The Langevin function is an odd function, L(−y)=−L(y), and in a neighborhood of zero can be expanded

as

L(y)=1

3y− 1

45y3 +

2

945y5 +O(y6). (2.59)

2.8 Diamagnetism, Paramagnetism and Ferromagnetism: a microscopic interpretation.

When a piece of matter is placed in a region of space in which a magnetic induction field is present, it is

subject to a mechanical action and simultaneously change the configuration of the field. When analyzing

the behavior of different kind of matter in a magnetic field, some phenomenological characteristics emerge

which allow to identify three families of substances.

Paramagnetic

Ferromagnetic

Diamagnetic

Figure 2.10. Some materials toward the inside of the solenoid: these kind of substances are called ferromagneticsubstances. Other substance are attracted by a force of many orders of magnitude lower than in the case of aferromagnetic substance: these are called paramagnetic substances. Finally, other substance are subject to forces ofthe same order of magnitude compared to the paramagnetic substance, but are rejected instead of being attracted:these substances are called diamagnetic substances.

For example, by partially inserting into a solenoid, traversed by a current, some cylindrical samples of

different materials, we immediately experiment that some materials, such as iron, cobalt and nickel, are

attracted by a very intense force (of the order of the weight force or more) toward the inside of the solenoid:

these kind of substances are called ferromagnetic substances. Other substance such as aluminum,

platinum and chromium, although still drawn towards the interior of the solenoid, however, are attracted

by a force of many orders of magnitude lower than in the case of a ferromagnetic substance: these are

called paramagnetic substances. Finally, other substance such as copper, lead, sulfur, carbon, and

silver are subject to forces of the same order of magnitude compared to the paramagnetic substance, but

are rejected instead of being attracted: these substances are called diamagnetic substances.

2.8.1 Diamagnetic materials

Diamagnetic materials can be microscopically characterized as those materials whose atoms have no

intrinsic magnetic moment. In them, the magnetic polarization is only due to Larmor precession. If we

denote by n the number of atoms per unit volume, taking into account the expression of the total Larmor

magnetic moment (2.56), from the defining equation of M expressed by (2.39), we get

M=−nZe2a2

6meBl=−nZe

2a2

6meµ0Hl. (2.60)


31

Substituting in the previous equation, the Lorentz local field formula (2.50) we get

M= χdH , (2.61)

where

χd :=− |αd|1+

1

3|αd|

and αd :=−nZe2a2

6meµ0. (2.62)

The quantity χd so defined is called the magnetic susceptibility of the material, and is a negative

quantity which does not depend on the temperature. Sinceχd

αd=

1

1 +1

3|αd|

, and since |αd|≪ 1, we get that

χd⋍αd. Typical numerical values of the susceptibility χd can be obtained by substituting typical valued

in the equation (2.62) for αd, and approximating χd with αd. For example by putting a2 = 10−20m2,

n = 5 · 1028m−3 and Z = 10 we finish with a (negative) value of χd which is of the order of 10−5 and

therefore in good qualitative agreement with the experimental data.

2.8.2 Paramagnetic materials.

In paramagnetic materials, although atoms have a non vanishing intrinsic magnetic moment, the value

of this intrinsic magnetic moments and the general structure of the paramagnetic material, do not allow

the presence of intense values of the local field Bl inside of the material: indeed, except at very low

temperatures (T → 0), the modulus of the magnetization remains very far from its saturation value.

Under these conditions (i.e. T not to low), the relationship between the local field Bl = µ0Hl and the

macroscopic fields M and H, can be assumed to satisfy the Lorentz local field formula (2.50) (as in the

case of diamagnetic materials). On the other hand, due to the presence of small values of the local field

Bl inside a paramagnetic materials, the constitutive relation between M = nm0 and the local field Bl,

can be taken as the first order approximation of the Langevin relation (2.57) in which L(y)=1

3y+O(y2).

We so finish with the following system of equations:

Hl = H+1

3M

M = αpHl

with αp :=n|〈m0〉|2µ0

3kBT. (2.63)

In writing the second of these equations we have taken into account that the average vector magnetic

moment 〈m0〉, whose form is given by (2.57), is directed as Hl. Combining the two previous relations,

we obtain:

M= χpHl (2.64)

where

χp :=|αp|

1− 1

3|αp|

. (2.65)

The quantity χp so defined is still called themagnetic susceptibility of the material, and no confusion

may arise since now χp is a positive quantity (and depends on the temperature T ). Since even in the

case of paramagnetic materials |αp|≪ 1 we get χp⋍ |αp|.


32

2.8.3 Ferromagnetic materials.

In the case of ferromagnetic specimens the relationship between the local field Bl and the macroscopic

fields M and H is the Weiss equation (2.52). Due to the high value assumed by the Weiss constant γ, the

local magnetic induction field can assume, for small values of M, an intensity of the order of 5 ·103 Tesla.

The relationship between M and Bl is given by Langevin relation (2.57) (multiplied by the number n of

atoms per unit volume). However, as a consequence of the high values that the local field Bl is allowed to

assume, the Langevin function L can no longer be approximated to the first order in the y variable, and

this means that the Langevin function L can now reach the saturation value even at room temperature.

The relations to be used to find the constitutive relationship M= G(H) are

M(y)=n(m0 · el)L(y)el

M=1

γ(Hl−H)

with y :=m0 ·Bl

kBT≡ µ0

m0 ·Hl

kBT. (2.66)

Remark 2.4. We note that in the two previous equations the magnetization M is a function of the point

x∈Ω. Actually they are only a concise form to write M(y(x)) and M(x).

Due to the definition of y and the fact that the average effective magnetic moment 〈m0〉 is in the

same direction of Hl, we can rewrite the previous system of equations as

M(y)=n(m0 · el)L(y)el

M(y)=kBT

γµ0(m0 · el)y− 1

γH .

(2.67)

The quantityMs=n(m0 ·el) is called the saturation value of the magnetization M, and it represents

the maximum value of the first of the two equations, and therefore the maximum value of intensity of

magnetization achievable.

The solutions of (2.67) can be investigated by standard methods. In particular it is immediate to recog-

nize the relationship between the macroscopic fields M and H is, in general, not a functional dependence:

indeed the solutions of that system of equations constitute a locus in the plane which reproduces quite

well the qualitative characteristics of the general hysteresis curves obtained experimentally. It is therefore

interesting to note that by treating the temperature as the parameter of a bifurcation investigation, there

is a critical value of the temperature Tc, known as Curie temperature, above which the system has

always only one solution, whatever the value of H. This critical value (of the temperature) can be found by

imposing the equality of the first derivative, in y=0, of both equations of the system (2.67). We thus found

Tc :=γµ0(m0 · el)Ms

3kB. (2.68)

Both Curie temperature Tc and of saturation value of magnetization Ms play an important role in

the phenomenological theory of ferromagnetism: indeed, from their direct experimental knowledge, it is

possible to get an indirect experimental knowledge of the Weiss constant γ:

γ=3nkBTcµ0Ms

2 . (2.69)


33

Summarizing, for values T > Tc the material does no long present hysteresis (whatever the value of

H), and no longer behaves as a ferromagnet: indeed it start to behaves like a paramagnetic material. A

quantitative physical interpretation of this paramagnetic character can be explained as follows: as far as

T > Tc, the parameter y remains small even for values of the H field varying in a wide range; therefore

the first equation of (2.67) can be approximated to first order around the value y= 0, and we get

M =1

3y(m0 · el)el = Tc

γTHl. (2.70)

Substituting in the previous relation the second equation of (2.67) we finish with

M= χpH (2.71)

with

χp :=Tc

γ(T −Tc)≡ µ0n(m0 · el)2

3kB

1

(T −Tc). (2.72)

We have so established the so called Curie second law which is usually written in the form

χp=Cρ

(T −Tc)with Cρ :=

µ0n(m0 · el)23kB

. (2.73)


34

3Volume and Surface Potentials

The origins of potential theory can be traced out from the studies related to the properties of forces

which follow the universal law of gravitation stated by Isaac Newton in his Philosophiae Naturalis

Principia Mathematica (1687). The statement of this law is limited to the case in which the forces of

mutual attraction act upon two material particles of small size or two material points: that is why, the

study of the forces of attraction of a material point by a finite smooth material body (in particular an

ellipsoid, since many celestial bodies have this shape), was one of the most important problems in the

mathematics of celestial mechanics for more than two centuries after Newton enunciated his law. After

first partial achievements by Newton and others, studies carried out by Lagrange (1773), Legendre

(1784-1794) and Laplace (1782-1799) became of major importance. Lagrange has established that a

field of gravitational forces, as it is called now, is a potential field and has introduced a function which

was later called by Green (1828) a potential function and by Gauss (1840) just a potential.

Already Gauss and his contemporaries discovered that the method of potentials can be applied not

only to solve problems in the theory of gravitation but, in general, to solve a wide range of problems in

mathematical physics arising from electrostatics, magnetostatics, geodesy, and elastomechanics. In this

connection, potentials became to be considered not only for the physically realistic problems concerning

the mutual attraction of positive masses, but also for problems with masses of arbitrary sign, or

charges. The principal boundary value problems were defined, such as the Dirichlet problem and the

Neumann problem, the electrostatic problem of the static distribution of charges on conductors or the

Robin problem.

To solve the above-mentioned problems in the case of domains with sufficiently smooth boundaries

certain types of potentials turned out to be efficient, i.e. special classes of parameter-dependent integrals

such as volume potentials of distributed mass, single and double layer potentials, logarithmic potentials,

Green potentials. Results obtained by A.M. Lyapunov and V.A. Steklov at the end of the 19th century

were fundamental for the creation of strong methods of solution of the principal boundary value problems.

Studies in potential theory concerning properties of different potentials have acquired an independent

significance.

In the first half of the 20th century, a great stimulus for the generalization of the principal problems

and the completion of the existing formulations in potential theory was made on the basis of the general

notions of a Radon measure, a capacity and generalized functions. Modern potential theory is closely

related in its development to the theory of analytic, harmonic and subharmonic functions and to proba-

bility theory.

35

In the last decades, incited by the rapidly increasing computational power the numerical treatment

of boundary value problems in potential theory has become an interesting field of present research. Two

powerful numerical methods namely the boundary element method and the finite element method have

been developed and successfully applied to various problems in engineering mathematics. The foundations

of these techniques such as Sobolev spaces and the concept of strong ellipticity are necessary for proper

understanding and working with these methods.

3.1 The Laplace Operator and the Poisson’s Equation.

This section is mainly devoted to fix some notations. We call the Laplace operator or the Laplacian

of dimension N , the linear differential operator with constant coefficients in RN

∆ :=∂2

∂x12 + ···+ ∂2

∂xN2 . (3.1)

From an elementary point of view, being given a point x∈RN and a function u defined in a neighborhood

of x, ∆u(x)=∂12u(x)+ ···+∂N

2 u(x), is defined under the condition of the existence of the partial derivatives

∂12u(x), ...,∂N

2 u(x). In this hypothesis one has ∆u(x)=div∇u(x) where for a filedm=(m1, ...,mN) defined

in a neighborhood of x, the divergence of m at x is defined by

divm(x)=∂m1

∂x1(x)+ ···+ ∂mN

∂xN, (3.2)

with the assumption that the partial derivatives ∂1m1(x), ..., ∂NmN(x) exist.

We now consider Ω an open set in RN. For every positive integer m ∈ N0 the laplacian ∆ maps

the space Cm+2(Ω) into the space Cm(Ω), in the sense in which for u ∈ Cm+2(Ω), the function ∆u:

x∈Ω 7→∆u(x) is of class Cm(Ω). Thus ∆ maps C∞(Ω) into itself.

Definition 3.1. Being given a function f ∈C0(Ω) defined in an open set Ω, we call Poisson’s equation

the partial differential equation

∆u= f on Ω. (3.3)

We say that u is a classical solution of Poisson’s equation (3.3), if u∈C2(Ω) satisfies

∆u(x) = f(x) ∀x∈Ω. (3.4)

We then call Laplace’s equation the homogeneous Poisson’s equation ∆u = 0 on Ω. Every classical

solution of the Laplace’s equation is called an harmonic function (in Ω).

Definition 3.2. Being given a distribution f on Ω, we call a distribution solution of Poisson’s equation

(3.3), each distribution u on Ω satisfying

〈u,∆ϕ〉= 〈f , ϕ〉 ∀ϕ∈D(Ω). (3.5)

Obviously for f ∈C0(Ω), u is a classical solution if and only if u∈C2(Ω) and is a distributional solution.

Volume and Surface Potentials

36

A fundamental property of the Laplace’s operator is its invariance under Euclidean transformations.

An Euclidean transformation is defined as an isometry of the euclidean distance in RN, that is to say, a

mapping ω of a part Ω of RN into RN such that

|ω(x)−ω(y)|= |x− y | ∀x, y ∈Ω (3.6)

It can be shown that an Euclidean transformation defined on a connected open set in RN is composed

of a translation and of an orthogonal transformation. We then have the following property, which is,

characteristic of the Laplacian:

Proposition 3.3. Let ω be an Euclidean transformation of Ω, f a distribution on Ω and u a distribution

solution of ∆u= f on Ω. Then the distribution v :=ωu defined as 〈ωu, ϕ〉 := 〈v, ϕω〉 is a distribution

solution of ∆v=ωf . In particular, if u∈C2(Ω) then ∆(u ω−1)= f ω−1=(∆u) ω−1.

We now introduce the following important notion:

Definition 3.4. A distribution E ∈ D ′(RN) is called an elementary solution (or fundamental

solution) of the Laplace operator ∆ if it satisfies the equation

−∆E= δ in D ′(RN). (3.7)

In other terms E ∈D ′(RN) is an elementary solution of ∆ when

〈E,∆ϕ〉=−〈δ, ϕ〉=−ϕ(0) ∀ϕ∈D(Ω). (3.8)

An elementary solution of the Laplace operator is not unique. Indeed if ∆E0 =0 in D ′(RN) then also

E+E0 is a fundamental solution for the Laplace operator. Obviously do not exist C∞(RN) elementary

solution due to the «singularity» in 0∈RN which every fundamental solution must take into account. In

this direction it is possible to show (motivated by the invariance of the Laplace operator under Euclidean

transformations) that [DiB10, Sch66]:

Proposition 3.5. If N >3, the locally integrable function E ∈Lloc1 (RN)∩C∞(RN\0) defined by the

position

E(x) := cN1

|x|N−2, cN :=

1

(N − 2)ωN, ωN := |SN−1|= 2π

N

2

Γ(

N

2

) , (3.9)

is the only C∞(RN\0) radially symmetric elementary solution of the Laplace operator ∆. This

particular fundamental solution is also referred to as the Newtonian kernel.

3.1.1 Green’s identities for bounded domains.

This section is devoted to recalling the classical divergence theorem (also know as integration by

parts formula) and its fundamental corollaries. Let Ω be a bounded open set in RN with boundary ∂Ω

of class C1. Then for every m of class C1(Ω)∫

Ω

divmdτ=

∫

∂Ω

m ·ndσ (3.10)

3.1 The Laplace Operator and the Poisson’s Equation.

37

where dτ is the Lebesgue measure in Ω and dσ denotes the Lebesgue surface measure on ∂Ω. By setting

m=u∇v with u∈C1(Ω), v∈C2(Ω) and taking into account the vector identity div(u∇v)=u∆v+∇u ·∇v,we immediately get the so-called Green’s first identity:

∫

Ω

[u∆v+∇u · ∇v ] dτ =

∫

∂Ω

u∂v

∂ndσ. (3.11)

If u, v are both in C2(Ω) then we can apply Green’s first identity to both functions, to get:

∫

Ω

[v∆u+∇u · ∇v ] dτ =

∫

∂Ω

v∂u

∂ndσ. (3.12)

Subtracting equations (3.11) and (3.12) we finish with the so-called Green’s second identity:

∫

Ω

[u∆v− v∆u ] dτ =

∫

∂Ω

(

u∂v

∂n− v

∂u

∂n

)

dσ. (3.13)

Remark 3.1. By setting u≡ 1 (resp. u≡ v) in we get that if v is harmonic in Ω then

∫

∂Ω

∂v

∂ndσ= 0

(

resp.

∫

Ω

|∇v |2 dτ =

∫

∂Ω

v∂v

∂ndσ

)

. (3.14)

Thus, if v is harmonic in a C1 open and connected set Ω, and such that ∂nv= 0 on ∂Ω (resp. v|∂Ω = 0),

then |∇v |2 =0 on Ω and therefore v is constant in Ω (reps. v= 0 in Ω).

3.1.2 Boundary value problems: uniqueness.

Let u∈C2(Ω) be a classical solution of the Poisson’s equation ∆u= f in Ω. If v ∈C2(Ω) is an harmonic

function, then obviously (since ∆ is a linear operator) ∆(u + v) = f . Thus, Poisson’s equation in Ω,

requires a suitable set of additional restraints to be well posed, such as prescribed conditions to satisfy

on the boundary ∂Ω of Ω.

Definition 3.6. Given f ∈C(Ω) and ϕ∈C(∂Ω), the Dirichlet problem for the operator ∆ in Ω consists

in finding a function u∈C2(Ω)∩C(Ω) satisfying

∆u= f in Ω, and u|∂Ω = ϕ on ∂Ω. (3.15)

Given ψ ∈C(∂Ω), the Neumann problem consists in finding a function u∈C2(Ω)∩C1(Ω) satisfying

∆u= f in Ω, and ∂nu= ψ on ∂Ω, (3.16)

where n denotes the outward unit normal to ∂Ω.

Using Green’s first identity it is simple to prove the following uniqueness result:

Proposition 3.7. Let Ω ⊆ RN be a C1 bounded and connected open set. There exists at most one

solution of the Dirichlet problem (3.15). In the case of the Neumann problem (3.16), two solution differ

by a constant.


38

Proof. If u1, u2∈C2(Ω)∩C(Ω) (resp. u1, u2∈C2(Ω)∩C1(Ω)) are two solution of the Dirichlet problem

(resp. of the Neumann problem), then by setting u :=u1−u2 we get that u is harmonic in Ω and satisfy

the homogeneous boundary condition u|∂Ω = 0 (resp. ∂nu= 0 on ∂Ω). With the position v := u Green’s

first identity gives ‖∇u‖Ω2 =0 (see Remark 3.1). Thus u is constant in Ω, and hence u1 =u2 + c for some

c∈R. This ends the proof for the Neumann problem. For the Dirichlet problem we simply observe that

0 = (u2− u1)|∂Ω = c.

Remark 3.2. From Remark 3.1, we get that when f=0, a necessary condition for the Neumann problem

(3.16) to be well-posed is 〈ψ〉Ω =0, where we have denote by 〈ψ〉Ω the integral average of ψ on Ω.

3.2 The Stokes identity on bounded and regular domains.

We have seen (in Proposition 3.5) that the function E: y 7→ cN |y |2−N, is a locally integrable function

in RN, whose distributional laplacian is equal (up to the sign) to the Dirac mass concentrated in zero:

∆E =−δ. In particular, due to Proposition 3.3, for every x∈RN one has ∆y[E(x− y)] =−δx. We now

perform an heuristic (Dirac style) argument: by the formal substitution v(y) :=E(x− y) in the Green’s

second identity (3.13), we get for every x∈Ω:

∫

Ω

u∆[E(x− y)]−E(x− y)∆u dτ =

∫

∂Ω

u∂n[E(x− y)]−E(x− y)∂nu dσ (3.17)

and therefore

−u(x)= 〈−δx, u〉 =

∫

Ω

E(x− y)∆u dτ +

∫

∂Ω

u∂n[E(x− y)]−E(x− y)∂nu dσ

= cN

∫

Ω

∆u

|x− y |N−2dτ + cN

∫

∂Ω

u∂n

[

1

|x− y |N−2

]

− ∂nu

|x− y |N−2dσ

(3.18)

It can be shown by a simple argument that the previous relation actually holds whenever u∈C2(Ω), and

it is known in literature as Stokes identity or third Green’s identity [DiB10]. The importance of

Stokes identity is in that it constitutes an implicit representation formula for smooth functions in Ω.

Note 3.8.We want to point out the attention on a possible misunderstanding. By the notation ∇E(x − y) we

mean the gradient of the function E evaluated in (x− y), id est

(∇E) (x− y) =

(

cN∇1

| · |N−2

)

(x− y) =

(

−1

ωN

·| · |N

)

(x− y) (3.19)

and is not an ambiguous one. On the other hand, the notation ∇(E (x− y)) or ∇(E(x− y)) is ambiguous until

we specify if the argument of the gradient is a function of x or a function of y. In other terms it is not clear if

we are considering ∇(E( · − y)) or ∇(E(x − · )). To remove this ambiguity it is usual in literature to denote by

∇x[E(x− y)] the function ∇[E( · − y)] and by ∇y[E(x− y)] the function ∇[E(x− · )]. It is useful to note that

∇x[E(x− y)] = ∇[E( · − y)]=∇E(x− y)∇y[E(x− y)] = ∇[E(x− · )]=−∇E(x− y)

(3.20)

3.2 The Stokes identity on bounded and regular domains.

39

and since ∇E does not depend on the sign of the argument, we get ∇x [E(x− y)] =−∇y[E(x− y)]. We finish byobserving that with the notation ∂n[E(x − y)] we mean that the gradient is computed with respect to the same

variable which evaluates n. For example ∂n(y)[E(x− y)]=∇y[E(x− y)] ·n(y) and therefore

∂n(y)[E(x− y)] =−∇x[E(x− y)] ·n(y) =−∇E(x− y) ·n(y). (3.21)

3.2.1 The Newtonian potential. The simple- and double-layer potentials.

It is customary to rearrange equation (3.18) as a sum of integral operators: u= −NΩ[∆u] + S∂Ω[∂nu]−D∂Ω[u], where for every x∈Ω,

NΩ[∆u](x) :=

∫

Ω

E(x− y)∆u dτ = cN

∫

Ω

∆u(y)

|x− y |N−2dτ ,

S∂Ω[∂nu](x) :=

∫

∂Ω

E(x− y) ∂nudσ = cN

∫

∂Ω

∂nu

|x− y |N−2dσ ,

D∂Ω[u](x) :=

∫

∂Ω

∂n[E(x− y)]udσ = cN

∫

∂Ω

u∂n

[

1

|x− y |N−2

]

dσ .

(3.22)

Due to the important role played in Potential Theory, these integral operators are referred to with a name:

NΩ[∆u] is the Newtonian potential defined on Ω by ∆u; S∂Ω[∂nu] is the simple-layer potential

defined by the function ∂nu on ∂Ω; D∂Ω[u] is the double-layer potential defined by the function u on

∂Ω. The simple and double layer potentials are often refereed as surface potentials.

Having acquired that if u∈C2(Ω) then

u=−NΩ[∆u] +S∂Ω[∂nu]−D∂Ω[u] in Ω, (3.23)

we want to investigate what happens outside Ω. To this end we observe that for every x ∈Ω′ = RN\Ω,

the function y∈Ω 7→E(x− y) is harmonic in Ω and therefore, applying the Green’s second identity we find

−NΩ[∆u] :=−∫

Ω

E(x− y)∆udτ

=

∫

∂Ω

u∂n[E(x− y)]−E(x− y)∂nudσ

= D∂Ω[u]−S∂Ω[∂nu].

(3.24)

Thus, the second member of equation (3.23), which is well defined also for x ∈ Ω′, is equal to zero for

every x ∈ Ω′. For this reason, for every x ∈ RN\∂Ω, the second member of equation (3.23) provides a

representation formula for the class Cχ2(Ω) of function u defined in R

N\∂Ω, such that u ∈ C2(Ω) and

u= 0 in Ω′. We summarize all this stuff in the following interior scalar representation formula

Theorem 3.9. Let Ω be a bounded and regular open set. If u∈C2(Ω) then

χΩu=−NΩ[∆u] +S∂Ω[∂nu]−D∂Ω[u] ∀x∈RN\∂Ω (3.25)

where we have denoted by χΩu∈Cχ2(Ω) the trivial extension (extension by zero) of u outside Ω.


40

In this section we have introduced the Newtonian potential and the surface potentials, but the meaning

and the main properties of these potentials will be investigate in later sections. For the time being we

just want to point out that surface potentials make it possible to convert a boundary value problem into

a boundary integral equation, and this kind of formulation can be obtained for more general operators

and more general problems as afar as a fundamental solution is known. Thus it constitutes a flexible

method with important implications. In particular, it is the theoretical basis for the boundary element

method, which may offer several advantages (in terms of cost) from the computational point of view in

numerical approximations, due to a dimension reduction [Sal10].

3.3 The Stoke identity on unbounded domains.

Boundary value problems in unbounded domains occur in many applications: for instance in magneto-

static problems, in capacity problems and in the scattering of acoustic or electromagnetic waves [Néd01].

As in the case of Poisson’s equation in all RN, a problem in an unbounded domain requires suitable

conditions at infinity to be well posed. For example, if B1 is the unit sphere of R3, the Dirichlet problem

∆u=0 in RN\B1 and u= 0 on ∂B1 has the family of solutions a(1− |x|−1) with a∈R. Therefore there

is no uniqueness. To restore uniqueness, a typical requirement is that u has a limit u∞ as |x|→∞. Given

a bounded domain Ω, we call the open exterior of Ω, the set Ω′ =RN\Ω. Without loss of generality

we will assume that 0∈Ω and we will only focus on problems defined on connected exterior sets, i.e. on

exterior domains. Note that ∂Ω′= ∂Ω.

Generally speaking, unicity results for problems in unbounded domains Ω′ can be reached as far as

the space of functions in which to look for a solutions are such that the Green’s identities hold on Ω′.

The next theorem is a step toward this direction:

Theorem 3.10. Let Ω be a bounded and regular domain. Let u,v∈C2(Ω′) such that |v(x)|∈O∞(|x|−pv)

with pv> 1. If |∇u(x)| ∈O∞(|x|−p∇u), |∇v(x)| ∈O∞(|x|−p∇v), |∆u(x)| ∈O∞(|x|−p∇u) and

p∇u>N − pv, p∆u> (N + 1)− pv, p∇v> pv+ 1, (3.26)

then the Green’s first identity for unbounded domains holds:

∫

Ω′

[v∆u+∇u · ∇v ] dτ =

∫

∂Ω′

v∂nu dσ. (3.27)

Moreover, if |∆v(x)| ∈O∞(|x|−p∇v), |u(x)| ∈O∞(|x|−pu) and

p∇u>max (N − pv, pu+1) , p∇v>max (N − pu, pv+1)p∆u> (N +1)− pv , p∆v> (N +1)− pu

(3.28)

then also the Green’s second identity for unbounded domains holds:

∫

Ω′

[u∆v− v∆u] dτ =

∫

∂Ω′

(u∂nv− v∂nu) dσ. (3.29)

Here we have denoted by n the outward unit normal vector to ∂Ω′, i.e. the one directed toward the

interior of Ω.


41

Note 3.11. If v =E is the fundamental solution of the laplacian, then pv =N − 2 and therefore the Green’s first

identity holds whenever p∇u > 2 and p∆u > 3. While a sufficient condition for the validity also of the Green’s

second identity is that pu =1, p∇u = 2 and p∆u = 3.

Proof. If u, v ∈C2(Ω′) then for every bounded and regular open set λΩ⊃Ω one has in Ωλ :=λΩ\Ω∫

Ωλ

[v∆u+∇u · ∇v ] dτ =

∫

∂Ω′

v∂nu dσ+

∫

∂[λΩ]

v∂nu dσ (3.30)

Hence, if (λn)n∈N is a strictly increasing sequence of real numbers converging to +∞ and such that

λ1Ω ⊃ Ω, then Ω′ can be covered by the family (λnΩ\Ω)n∈N = (Ωλn)n∈N, id est Ω′ =

⋃

n∈NΩλn

(see

Figure 3.1). Moreover, for every n∈N

∫

Ωλn

[v∆u+∇u · ∇v] dτ =

∫

∂Ω′

v∂nu dσ+

∫

∂[λnΩ]

v∂nudσ. (3.31)

To conclude the proof it is sufficient to prove that

limn→∞

∫

Ωλn

[v∆u+∇u · ∇v] dτ =

∫

Ω′

[v∆u+∇u · ∇v] dτ (3.32)

and

limn→∞

∫

∂[λnΩ]

v∂nu dσ=0. (3.33)

We start by proving (3.32). In doing this we look for the values of the positive integers pv, p∆u, p∇u, p∇vsuch that the result holds whenever |v(x)| ∈O∞(|x|−pv), |∆u(x)| ∈O∞(|x|−p∆u), |∇u(x)| ∈O∞(|x|−p∇u)

and |∇v(x)| ∈O∞(|x|−p∇v). To this end we observe that for sufficiently large n∈N

(|v∆u|+ |∇u · ∇v |)χΩλn6

(

c(v,∆u)1

|x|pv+p∆u+ c(∇u,∇v)

1

|x|p∇u+p∇v

)

χΩ′ , (3.34)

where we have denoted by c(·,·) the constants arising from the respective O∞ class. Therefore if pv+ p∆u>

N and p∇u+ p∇v>N from Lebesgue dominated convergence theorem we get (3.32).

We now focus on the proof of (3.33). To this end we observe that for sufficiently large n∈N

∣

∣

∣

∣

∣

∫

∂[λnΩ]

v∂nudσ

∣

∣

∣

∣

∣

6 λnN−1

∫

∂Ω

|v(λny)| |∇u(λny)| dσ

6 c(v,∇u)λnN−1

|λn|pv+p∇u

∫

∂Ω

1

|y |pv+p∇udσ.

(3.35)

The last member of the previous relation converges (for n→ ∞) to zero as far as pv + p∇u > N − 1.

Therefore Green’s first formula is valid as far as

pv+ p∇u>N, pv+ p∆u>N + 1, p∇u+ p∇v>N + 1. (3.36)


42

By assuming the validity of the previous expressions for pv> 1 we see that necessarily

p∇u>N − pv, p∆u> (N +1)− pv, p∇v> pv+1. (3.37)

Interchanging the role of u and v, and assuming the validity of the previous expressions for pv>1 we see

that necessarily

p∇v>N − pu, p∆v> (N + 1)− pu, p∇u> pu+1. (3.38)

We thus arrive at the conclusion that once assigned pu, pv>1 the Green’s second identity is valid as far as

p∇u>max (N − pv, pu+1) , p∇v>max (N − pu, pv+ 1) (3.39)

p∆u> (N + 1)− pv , p∆v> (N + 1)− pu. (3.40)

and this ends the proof.

To extend Stokes’ identity (Green’s third identity) to an exterior region, a sufficient condition to

impose is the decay condition: |∆u| ∈O∞(|x|−3). More precisely:

Theorem 3.12. Let Ω be a bounded and regular domain. If u∈C2(Ω′) and

|u(x)| ∈O∞(|y |−1) , |∇u| ∈O∞(|y |−2) , |∆u| ∈O∞(|y |−3) (3.41)

then the Green’s third identity for unbounded domains holds:

u=−NΩ′[∆u] +S∂Ω′[∂nu]−D∂Ω′[u] in Ω′ . (3.42)

Note 3.13. Even though from a set theoretical point of view ∂Ω=∂Ω′, since the surface potentials are defined on

oriented boundaries, the normal vector field appearing in the expression of S∂Ω′[∂nu] relative to the unbounded

domain Ω′, is the opposite of the simple layer potential S∂Ω[∂nu] appearing in the interior scalar representation

formula. The same think holds for the double layer potential. In symbols:

S∂Ω′[∂nu] =−S∂Ω[∂nu] , D∂Ω ′[u] =−D∂Ω[ϕ]. (3.43)

In fact the convention adopted since now, has been to denote always with n the normal field inside the integral sign,

letting the domain of integration (∂Ω or ∂Ω′) to be a mark for in what direction is oriented the normal vector field.

Note 3.14. In this theorem appear the classes O∞(|y |−1), O∞(|y |−2) and O∞(|y |−3), is a consequence of what

said in Note 3.11.

Proof. If u∈C2(Ω′∪ ∂Ω) then for every spherical region BR of radius R such that BR⊃Ω, one has in

BR\Ω the classical Stokes’ identity for bounded and regular domains

u(x) =S∂Ω′(∂nu) +D∂Ω′(−u)+NBR\Ω(−∆u)+S∂BR(∂nu)+D∂BR

(−u) ∀x∈BR\Ω. (3.44)


43

On the other hand, for every x∈BR\Ω, since E(x− y) is harmonic in BR′ , due to Green’s second identity

applied to BR′ , we have

NBR′ (−∆u) =D∂BR

′ (u)−S∂BR′ (∂nu) = D∂BR

(−u)+S∂BR(∂nu), (3.45)

the last equality following from the fact that ∂BR′ =−∂BR. Substituting the previous equality in (3.44)

we get

u(x)=S∂Ω′(∂nu) +D∂Ω′(−u)+NBR\Ω(−∆u)+NBR′ (−∆u) ∀x∈BR\Ω. (3.46)

We thus have proved that for every R> 0 such that BR⊇Ω

u(x)= NΩ′(−∆u)+S∂Ω′(∂nu)+D∂Ω′(−u) ∀x∈BR\Ω . (3.47)

and since R is arbitrary, the proof is concluded.

Having acquired that if u∈C2(Ω′), u∈O∞(|y |−1),∇u∈O∞(|y |−2) and ∆u∈O∞(|y |−3)then

u=NΩ′[−∆u] +S∂Ω′[∂nu] +D∂Ω′[−u] in Ω′ (3.48)

we want to observe that for every x∈Ω the function y ∈Ω′ 7→E(x− y) is harmonic in Ω and therefore,

applying the Green’s second identity (see Note 3.11) we get NΩ′[−∆u]=D∂Ω′[u]−S∂Ω′[∂nu] and therefore

the second member of equation (3.48), which has a meaning also for x∈Ω, is equal to zero for every x∈Ω.

For this reason, for every x ∈ RN\∂Ω, the second member of equation (3.48) furnish a representation

formula for every function u defined in RN\∂Ω such that u∈C2(Ω′) and u= 0 in Ω. We summarize all

this stuff in the following:

Theorem 3.15. Let Ω be a bounded and regular open set. If u∈C2(Ω′), u∈O∞(|y |−1),∇u∈O∞(|y |−2)

and ∆u∈O∞(|y |−3) then

χΩ′u=−NΩ′[∆u] +S∂Ω′[∂nu]−D∂Ω′[u] in RN\∂Ω (3.49)

where we have denoted by χΩ′u∈Cχ2(Ω′) the trivial extension (extension by zero) of u inside Ω.

3.3.1 Maximum principles on exterior domains.

Maximum principles play a fundamental and unifying role in the proof of uniqueness results concerning

boundary value problems for elliptic equations. We refer to [DiB10, ABW01] for classical results con-

cerning maximum principles on bounded domain. Here we will focus on the exterior domains results (see

[Sal10]).

Proposition 3.16. Let Ω′⊆RN be an exterior domain. Let u∈C2(Ω′)∩C(Ω′) be an harmonic function

in Ω′ vanishing at infinity, i.e. such that

limx→∞

u(x) =0. (3.50)

If u> 0 (resp. u6 0) on ∂Ω′ then u> 0 (resp. u6 0) in Ω′.


44

Note 3.17. The proof is based on the analog result for bounded domains. To this end we recall (see [Sal10]) that

if Ω is a bounded domain, u∈C2(Ω)∩C(Ω) is harmonic in Ω and u > c on ∂Ω for some c∈R, then u > c in Ω.

Proof. Let u> 0 on ∂Ω′. Since limx→∞u(x) = 0, for every ε > 0 there exists rε∈R+ such that Ω⊂Br

and u>−ε in ∂Br for every r> rε. Therefore once considered the bounded domain Ωr :=Ω′∩Br, we haveu>−ε on ∂Br and on ∂Ω′, and hence u>−ε in ∂Ωr. It then follows that for every ε>0 and every r> rε

u>−ε in Ωr, (3.51)

where rε can be chosen such that limε→0+ rε = +∞. By taking the limit for ε → 0+ of the previous

inequality we reach the desired result.

We are now able to prove the following fundamental result concerning the behavior around infinity

of harmonic functions vanishing at infinity, id est such that limx→∞u(x)= 0:

Theorem 3.18. Let Ω be a bounded and regular domain. If u∈C2(Ω′) is such that

∆u=0 in Ω′ and limx→∞

u(x)= 0 (3.52)

then

|u(x)| ∈O∞(|y |2−N), |∇u(x)| ∈O∞(|y |1−N), |∂iju(x)| ∈O∞(|y |−N). (3.53)

Therefore (for N >3) the previous Green’s identities hold. In particular we have the following exterior

scalar representation formula for functions harmonic in Ω′:

χΩ′u=S∂Ω′[∂nu]−D∂Ω′[u] in RN\∂Ω. (3.54)

Proof. We prove only the fist decay condition |u(x)|∈O∞(|y |2−N). We refer to [Sal10] for the other ones.

Since Ω is bounded there exists an rΩ>1 such that |y |>rΩ⊆Ω′. Since u vanishes at infinity, there exists

an ru∈R+ such that |u(y)|61 for |y |> ru. Next we pose r0 :=maxrΩ, ru and we note that the function

w(y) :=u(y)− r0N−2

|y |N −2is harmonic for |y |>r0, is non positive on |y |=r0 and vanishes at infinity: in symbols

∆w(y)= 0 for |y |> r0, w(y)6 0 for |y |= r0, limy→∞

w(y)= 0. (3.55)

Therefore, by Proposition 3.16,

w(y)6 0 in Ω′∩ |y |> r0. (3.56)

By setting v(y) :=−u(y)− r0N −2

|y|N −2, we still have

∆v(y)= 0 for |y |> r0, v(y) 6 0 for |y |= r0, limy→∞

v(y)= 0. (3.57)

and therefore

v(y)6 0 in Ω′∩|y |> r0. (3.58)


45

From the two previous relation (3.56) and (3.56), we so get |u(y)| 6 r0N−2|y |2−N for y ∈ Ω′ ∩ |y |> r0

and therefore |u(y)| ∈O∞(|y |2−N).

From the scalar representation formulas for bounded and unbounded domains (Theorems 15 and 18),

we immediately obtain the following fundamental jump scalar representation theorem:

Corollary 3.19. Let Ω be a bounded and regular domain of RN . For a scalar function

χΩui+ χΩ′u2 =:u∈C1(Ω)∩C1(Ω′)∩C2(RN\∂Ω) (3.59)

such that, ∆u=0 in Ω′ and limx→∞u(x)=0 the following jump scalar representation formula holds:

u(x)= cN

−∫

Ω

∆u(y)

|x− y |N−2dµ−

∫

∂Ω

(∂nue− ∂nui)(y)

|x− y |N−2dσ

+

∫

∂Ω

(ue− ui)(y)∂n

[

1

|x− y |N−2

]

dσ

(3.60)

for ever x∈RN\∂Ω. Equivalently:

u=NΩ[−∆u] +S∂Ω[−(∂nue− ∂nui)] +D∂Ω[ue−ui] in RN\∂Ω. (3.61)

The scalar function u is so decomposed into a single layer potential of the volume density −∆u in Ω, a

single layer potential of the surface density −(∂nue− ∂nui) on ∂Ω and a double layer potential of the

surface density ue− ui on ∂Ω.

Note 3.20. It is customary to refer to the quantity ue −ui defined on ∂Ω as the jump of the function u along∂Ω, and to denote it with the symbol u −

+ or u ie. The same convention applies for the jump of the normal

derivative ∂nue − ∂nui. Therefore:

u −+ :=ue − ui , ∂nu −

+ := ∂nue − ∂nui for y ∈ ∂Ω. (3.62)

With this new notation equation (3.61) reads as

u =NΩ[−∆u] +S∂Ω[−∂nu −+] +D∂Ω[u −

+ ] in RN\∂Ω. (3.63)

Even if obvious, it is important to observe that the jumps u −+ and ∂nu −

+ are continuous functions on ∂Ω(being the difference of two continuously differentiable functions on ∂Ω).

Proof. We denote by ui and ue the continuous extensions of the C1(Ω) part of u and of the C1(Ω′) part

of u. From the Stokes identity for bounded and unbounded regular domains we respectively get (see

Theorem 15)

χΩui=NΩ[−∆ui] +S∂Ω[∂nui]−D∂Ω[ui] in RN\∂Ω ,

χΩ′ue=S∂Ω′[∂nue]−D∂Ω′[ue] in RN\∂Ω .

(3.64)

But (cfr. note 13) since S∂Ω′[∂nue] =−S∂Ω[∂nue] and D∂Ω′[−ue] =D∂Ω[ue], we have

u= χΩui+ χΩ′ue=NΩ[−∆ui] +S∂Ω[∂nui]−D∂Ω[ui]−S∂Ω[∂nue] +D∂Ω[ue]. (3.65)

Grouping the terms inside the same surface potentials completes the proof.


46

3.4 Surface potentials

Given a bounded and regular domain Ω (say of class C2), as we have already seen, the double layer

potential is defined for every u∈C(∂Ω) by

D∂Ω[u] =

∫

∂Ω

u(y)∂n[E(x− y)] dσ. (3.66)

From a physical point of view, in three dimensions, it represents the electrostatic potential generated by

an ideal dipole potential distribution of moment u on ∂Ω (Figure 3.1).

∂Ωu(y)

Figure 3.1. From a physical point of view, in three dimensions, it represents the electrostatic potential generatedby an ideal dipole potential distribution of moment u on ∂Ω.

To better understand what we mean, we recall that if y∈∂Ω and −q(y) and q(y+εn) are two charges

placed at points y and y+ εn, we say that the pair of charges (−q(y), q(y+ εn)), with q >0, constitutes

a physical dipole of axis n(y) and dipole moment uε given by the product of the charge modulus

with the vector distance εn(y)

uε(y) := q(y) εn(y). (3.67)

The induced potential at a point x∈R3\y, y+ εn is well known and given (up to a constant) by

vε(x, y) =−q(y)[E(x− (y+ εn))−E(x− y)]. (3.68)

We then define the ideal dipole potential as the limiting distribution obtained by letting ε→0 with the

constraint that the modulus uε(y) := |uε(y)|= |εq(y)| of the moment of dipole remains constant, id est by

letting the function q(y), for fixed y∈∂Ω, now to be a function also of ε and such that εqε(y) :=u(y)∈R+

for some positive constant u(y) and for every ε in a positive neighborhood of zero. We can then rewrite

the previous equation in the form

vε(x, y) =−u(y)E(x− (y+ εn(y)))−E(x− y)

ε, (3.69)

and letting ε go to zero we finish with

v(x, y) := limε→0

vε(x, y)=−u(y)∂n(y)E(x− y). (3.70)

Integrating on ∂Ω we obtain, up to a negative constant, exactly D∂Ω[u]. This is the reason why some

author prefer to define the double layer potential as the opposite (in sign) of the one here defined.


47

The study of the main properties of DΩ, as often in science, can be deduced starting from more simple

situations. In this respect, particularly useful is to look at the case u(y) ≡ 1 on ∂Ω, i.e. to look at the

operator

D∂Ω[1] =

∫

∂Ω

∂n[E(x− y)] dσ= cN

∫

∂Ω

∂n

[

1

|x− y |N−2

]

dσ (3.71)

Inserting u ≡ 1 into the interior scalar representation formula (3.25), we get D∂Ω[1](x) = −χΩ(x) ∀x ∈RN\∂Ω. The situation become more complex (and interesting) whenever x∈∂Ω. In this case it is possible

to show the following (see [DL00, Sal10]):

Lemma 3.21. If Ω is a bounded C2-domain of RN then

D∂Ω[1](x) =−(

χΩ(x)+1

2χ∂Ω(x)

)

≡

−1 if x∈Ω

−1

2if x∈ ∂Ω

0 if x∈Ω′

(3.72)

Thus, when u≡1, the double layer potential is piecewise constant outside ∂Ω and has a jump discon-

tinuity across ∂Ω.

Note 3.22.Observe that if x∈ ∂Ω then

limz→x

z∈Ω

D∂Ω[1](z)=D∂Ω[1](x)−1

2, lim

z→x

z∈Ω ′

D∂Ω[1](z) =D∂Ω[1](x) +1

2(3.73)

In fact the previous to relations are only a complicated way of writing the relations −1=−1

2− 1

2and 0=−1

2+

1

2.

Despite of their simplicity, the previous observation (and limiting relations) are the key point in

understanding the general properties of D∂Ω[u] stated in the following (see [DL00, Sal10, Lay08]):

Theorem 3.23. Let Ω⊆RN be a bounded C2-domain and u∈C(∂Ω). Then the double layer potential

D∂Ω[u] is harmonic outside ∂Ω (i.e. inRN\∂Ω) and the following jump relations hold for every x∈∂Ω:

limz→x

z∈Ω

D∂Ω[u](z)=D∂Ω[u](x)− 1

2u(x) ,

limz→x

z∈Ω′

D∂Ω[u](z)=D∂Ω[u](x)+1

2u(x) .

(3.74)

Given a bounded and regular domain Ω, as we have already seen, the single layer potential is defined

for every u∈C(∂Ω) by

S∂Ω[u] =

∫

∂Ω

u(y)E(x− y) dσ. (3.75)

From a physical point of view, in three dimensions, it represents the electrostatic potential generated by

a charge distribution of surface density u on ∂Ω. The main results concerning the single layer potential

are condensed in the following (see [DL00, Sal10, Lay08]):


48

Proposition 3.24. Let Ω⊆RN be a bounded C2-domain and u∈C(∂Ω). Then S∂Ω[u] is continuous in

RN and harmonic in RN\∂Ω. Moreover denoted by n a continuous extension of n to RN, the function

∇S∂Ω[u] · n defined in RN\∂Ω can be extended from the inside of Ω to Ω, from the outside of Ω to Ω′,

and following jump relations hold for every x∈ ∂Ω

limz→x

z∈Ω

∂n(z)S∂Ω[u](z)=

∫

∂Ω

u(y)∇E(x− y) ·n(x)dσ +1

2u(x) ,

limz→x

z∈Ω′

∂n(z)S∂Ω[u](z)=

∫

∂Ω

u(y)∇E(x− y) ·n(x)dσ − 1

2u(x) .

(3.76)

and therefore ∂n(z)S∂Ω[u] −+ =−u.

Remark 3.3. It is important to distinguish clearly between the function

∫

∂Ω

u(y)∇E(x− y) ·n(x)dσ (3.77)

appearing in the expression of the normal derivative of the simple layer potential, and the function

D∂Ω[u](x)=

∫

∂Ω

u(y)∂nE(x− y) dσ=

∫

∂Ω

u(y)∇E(x− y) ·n(y) dσ (3.78)

defined by the double layer potential. These two functions are related by the equation

∫

∂Ω

u(y)∇E(x− y) ·n(x)dσ−D∂Ω[u](x)=

∫

∂Ω

u(y)∇E(x− y) · (n(x)−n(y)) dσ. (3.79)

We conclude this section by enunciating an important result concerning the regularity of the derivative

of interior and exterior simple layer potentials right up to the boundary (cfr. [DL00]):

Proposition 3.25. If ∂Ω is of class Cm+1,α and u ∈ Cm,α(∂Ω) with m ∈N and 0 < α < 1, then the

interior (resp. exterior) simple layer potential S∂Ω[u] defined by u is of class Cm+1,α on Ω (resp. on Ω′).

Similarly the interior (resp. exterior) double layer potential D∂Ω[u] is of class Cm,α on Ω (resp. on Ω′).


49

4The Demagnetizing Field

In this chapter we mainly focus our attention to the case N =3. Objective of this chapter is to apply the

results of the previous chapter to the study of the main properties of the demagnetizing field. We start

by recalling some regularity properties of the Newtonian potential.

4.1 The Newtonian Potential. Regularity.

From the third Green’s identity many interesting consequences can be deduced. To this end we start by

recalling some basic facts about convolution. For measurable functions u, v ∈R3 →R3 we define, the

convolution u⋆v of u and v by

(u⋆v)(x) :=

∫

R3

u(x− y)v(y) dy, (4.1)

for all x∈R3 for which the integral exists. As usual in the context of convolutions, measurable functions

u defined in a subset Ω of R3, are identified with their trivial extension

u(x) :=

u(x) if x∈Ω

0 if x∈ ∁Ω .(4.2)

We summarize some well-known facts about convolutions (cfr. [Bre10]):

Proposition 4.1. For 1 6 p6∞ and k ∈N, if u∈Llocp (R3) and v ∈Cck(R3) then we have

u⋆v ∈Ck(R3) with ∂α(u⋆v)= u⋆ (∂αv) (4.3)

for any multi-index α∈Nk3 with |α|=∑

i=13

αi6 k.

The Newtonian kernel, for N=3, reads as E(x)=(4π)−1|x|−1. As already pointed out E is in Lloc1 (R3)

and therefore, as an immediate consequence of the previous theorem, for every function f ∈Cc∞(R3) the

convolution E ⋆ f is well defined (and belongs to C∞(R3)). Moreover the Newtonian potential of f can

be conveniently expressed in terms of convolution as

N [f ] =E ⋆ f ∀f ∈Cc∞(R3). (4.4)

51

Since the Newtonian kernel is the fundamental solution of the Laplace operator, and the δ distribution is

the identity element in the algebra of convolutions, we immediately get −∆N [f ]=−∆E ⋆ f =(δ ⋆ f)= f .

In other terms, u :=N [f ] is a distributional solution of the Poisson equation −∆u= f (cfr. [Sch66]). Of

course the same conclusion can be reached in a classical setting [GT01] as an immediate consequence of

Proposition 4.1:

Proposition 4.2. For any function f ∈ Cc∞(R3) the Newtonian potential N [f ] := E ⋆ f is of class

C∞(R3) and satisfies (pointwise) the Poisson equation:

−∆N [f ] = f in R3. (4.5)

Furthermore, for the partial derivatives of N [f ] the following relations hold

∂jN [f ] =E ⋆ (∂jf). (4.6)

Proof. Indeed from the interior scalar representation formula (3.25) we get (in particular) that if f ∈Cc

∞(RN) then

f(x)= cN

∫

R3

−∆f(y)

|x− y |N−2dτ =−(∆f ⋆E). (4.7)

Since E ∈Lloc1 (RN), due to Proposition 4.1, (f ⋆E)∈C∞(R3) and −(∆f ⋆E) =−∆(f ⋆E). Hence:

f(x) =−∆

(

cN

∫

Ω

f(y)

|x− y |N−2dτ

)

=−∆N [f ] (4.8)

and this concludes the proof.

Let us now observe that if u ∈ C∞(Ω) with ∂Ω smooth, there exists an extension of u to the class

Cc∞(R3), that we denote by f := uχΩ +u1χΩ′ . Since f ∈Cc∞(R3) from Proposition 4.2 we get

uχΩ +u1χΩ′ = f = −∆N [f ] = −∆NΩ[u]−∆NΩ′[u1]. (4.9)

In particular, for every x∈Ω

u=−∆NΩ[u]−∆

[∫

Ω′

u1(y)E(x− y) dτ

]

. (4.10)

Differentiating under the integral sign, noting that E(x− y) is harmonic in Ω′ as far as the pole x is in

Ω, we finish with the following fundamental result, that we state in the vector setting:

Theorem 4.3. Let Ω be a bounded and smooth domain of R3. If m ∈ C∞(Ω, R3) and m(x) = 0 in

Ω′=R3\Ω then NΩ[m]∈C∞(Ω,R3)∩C∞(Ω′,R3) and

−∆NΩ[m] =m(x) in Ω∆NΩ[m] = 0 in Ω′ .

(4.11)

In other terms: the potential NΩ[m] is harmonic in Ω′; the function u := NΩ[m] is a solution of the

Poisson equation −∆u=m in Ω.

The Demagnetizing Field

52

More generally the following regularity result can be proved (cfr. [DL00]):

Theorem 4.4. If m is in the Holder class Cm,α(Ω), in particular if m∈Cm+1(Ω), then the Newtonian

potentialNΩ[m] is defined for all x∈RN and NΩ[m]∈C1(RN)∩Cm+2,α(Ω)∩C∞(Ω′). Moreover NΩ[m]

is harmonic in Ω′ (∆NΩ[m] = 0 in Ω′) and satisfies the Poisson equation −∆NΩ[m] =m(x) in Ω. In

other terms

χΩm=−∆NΩ[m] in R3\∂Ω (4.12)

If N > 3 the potential vanishes at infinity, id est:

limx→∞

NΩ[m](x)= 0. (4.13)

4.2 The Helmholtz-Hodge decomposition formula

In this section we investigate the resolvability of the following problem: given a vector field m in R3,

we want to find a decomposition of m into the sum of a divergence free and curl free vector field.

Precisely, given m we want to find a scalar function um and a vector filed Am such that the following

Helmholtz-Hodge decomposition formula (also called the fundamental theorem of vector

analysis) holds: m=∇um + curlAm.

To this end we assume that Ω is a bounded and smooth domain of R3, that m ∈ C∞(Ω) and

m(x) = 0 in Ω′ := R3\Ω. Since m ∈ C∞(Ω) we know, from Theorem 4.4, that NΩ[m] ∈ C∞(Ω)

and that it satisfies the Poisson equation −∆ NΩ[m] = m in R3\∂Ω. From the pure vector identity

−∆NΩ[m] = curl curlNΩ[m]−∇divNΩ[m], we get the desired result:

m = curlAm +∇um in R3\∂Ω. (4.14)

The quantities um :=−divNΩ[m] andAm :=curlNΩ[m], known in the physical literature respectively as

the magnetostatic scalar potential and the magnetostatic vector potential, are both in C0(R3)∩C∞(R3\∂Ω) due to Theorem 4.4, id est continuous in all R3 and smooth outside the boundary of Ω.

In this physical context, the vector m is called the magnetization vector associated to the physical

magnetic body occupying the region Ω.

4.2.1 The magnetostatic scalar and vector potentials. Integral representations.

We know look for a more explicit expression for the potentials um and Am. To this end we show two

possible roads to follow. The first one make use of the jump scalar representation formula (3.61), from

which we get (since um∈C0(R3) and therefore um+ − um

− = 0 on ∂Ω):

um =−NΩ[∆um] +S∂Ω[∂num− − ∂num

+ ] ∀x∈R3. (4.15)

On the other hand, from (4.14) we get

∆um =divm , ∂num− =m ·n− (curlAm)− ·n , ∂num

+ =−(curlAm)+ ·n, (4.16)


53

and it is possible to show that (curlAm)− ·n= (curlAm)+ ·n on ∂Ω. Thus we finish with

um =−NΩ[divm] +S∂Ω[m ·n] ∀x∈R3. (4.17)

The second possible road to follow for the computation of um is the direct one. Once recalled the vector

identity div (ϕm) = ϕdivm + ∇ϕ · m, and that since the Newtonian potential has a variable weak

singularity, it is possible to differentiate under the integral sign, for every x∈R3 we get:

um :=−divNΩ[m] =− 1

4π

∫

Ω

divx

[

m

|x− y |

]

dτ

=− 1

4π

∫

Ω

m(y) · ∇x

[

1

|x− y |

]

dτ =1

4π

∫

Ω

m(y) · ∇y

[

1

|x− y |

]

dτ

=1

4π

(∫

Ω

divy

[

m

|x− y |

]

dτ −∫

Ω

divm(y)

|x− y | dτ

)

=1

4π

(∫

∂Ω

m(y) ·n(y)

|x− y | dσy−∫

Ω

divm(y)

|x− y | dτ

)

,

(4.18)

and the previous relation, in terms of the Newtonian and simple layer potentials, reads as (4.17).

For the computation concerning Am it is still possible to follow the previous two roads. Indeed, by

making use of the jump scalar representation formula (3.61), observing that Am∈C0(R3) and therefore

Am+ −Am

− = 0 on ∂Ω, we get

Am =−NΩ[∆Am] +S∂Ω[∂nAm− − ∂nAm

+ ] ∀x∈R3. (4.19)

On the other hand, from (4.14) we get (observing that divAm =div curlAm =0)

∆Am =−curlm , ∂nAm− = ∂n(curlm)− , ∂nAm

+ = 0, (4.20)

and it is possible to show that ∂nAm− =m×n on ∂Ω. Thus we finish with

Am =NΩ[curlm] +S∂Ω[m×n] ∀x∈R3. (4.21)

The second possible road to follow for the computation of Am is the direct one. Once recalled the vector

identity curl (ϕv) = ϕ curlm+∇ϕ×m, and again that, since the Newtonian potential has a variable

weak singularity, it is possible to differentiate under the integral sign, we get for every x∈R3:

Am := curlNΩ[m] =1

4π

∫

Ω

curlx

[

m(y)

|x− y |

]

dτ

=1

4π

∫

Ω

∇x

[

1

|x− y |

]

×m(y) dτ =− 1

4π

∫

Ω

∇y

[

1

|x− y |

]

×m(y)dτ

=1

4π

(∫

Ω

curlm(y)

|x− y | dτ −∫

Ω

curly

[

m(y)

|x− y |

]

dτ

)

=1

4π

(∫

Ω

curlm(y)

|x− y | dτ +

∫

∂Ω

m(y)×n(y)

|x− y | dσy

)

(4.22)

and the previous relation, in terms on Newtonian and simple layer potential, reads as (4.21).


54

From the physical point of view, by (4.14), the vector field m is decomposed into an circulation

free part and a flux free part which originated by the Newton-Coulomb law for the source (charge)

distributions:

ρm :=−divm in Ω , σm :=m ·n on ∂Ω, (4.23)

and the Biot-Savart law for vortex (current) distributions

jm := curlm in Ω , km :=m×n on ∂Ω. (4.24)

If m is an harmonic vector field, i.e. a vector field such that jm :=curlm=0 in Ω and ρm :=−divm=0

in Ω then the potentials (4.17) and (4.21) reduce to surface integrals. In this case the Helmholtz-Hodge

decomposition formula is called Cauchy’s integral formula: indeed it is possible to show that in

dimension two the formula reduces to the classical Cauchy integral formula of Complex Analysis. We

summarize all the previous results in the following:

Proposition 4.5. Let Ω be a bounded and smooth domain of R3. If m ∈ C∞(Ω) and m(x) = 0 in

Ω′ :=R3\Ω then the Helmholtz-Hodge decomposition in terms of the magnetostatic scalar potential um

and the scalar vector potential Am holds:

m=b[m]−h[m] = curlAm +∇um in R3\∂Ω, (4.25)

where we have denoted by b[m] := curl Am and h[m] := −∇um the magnetic flux density field

and the demagnetizing field. These fields are both smooth outside the boundary: b[m], h[m] ∈C∞(R3\∂Ω), while the magnetostatic scalar potential um and the magnetostatic vector poten-

tial Am, are both continuous in all space and smooth outside the boundary:

um,Am∈C0(R3)∩C∞(R3\∂Ω). (4.26)

Moreover it is possible to express these fields as a sum of a Newtonian potential and a simple-layer

potential:

um = NΩ[ρm] +S∂Ω[σm] = NΩ[−divm] +S∂Ω[m ·n],Am = NΩ[jm] +S∂Ω[km] = NΩ[curlm] +S∂Ω[m×n],

(4.27)

where ρm := −divm and σm :=m · n are the volume and surface source charge distributions,

jm := curlm and km :=m×n are the volume and surface vortex current distributions.

Remark 4.1. We want to underline the complementary properties of magnetic flux density field and the

magnetic field: the magnetic field is curl free while the divergence is determined by the given function

(density of mass); the magnetic flux density field is divergence free while its curl is determined by the

given function (current density). The magnetic field is a gradient field but not a curl field (it has a scalar

potential but not a vector potential); the magnetic flux filed is a curl field and not a gradient field (it has

a vector potential, but not a scalar potential).

4.2.2 Transmission conditions for the magnetic flux density field b and the demagnetizing field h.

We have seen that the decomposition of m in terms of the demagnetizing field h[m] and the magnetic

flux density field b[m], is valid outside the boundary ∂Ω of Ω:

m=b[m]−h[m] = curlAm +∇um in R3\∂Ω. (4.28)


55

It is thus a natural question to investigate the behavior of these fields near the boundary ∂Ω. To this

end it is convenient to work with the integral representations of the potentials expressed in Proposition

4.5. Indeed from (4.25) and the obvious fact that m ·n −+ = 0−m ·n=−m ·n we get

b[m] ·n −+ =m ·n −

+ +h[m] ·n −+ =−m ·n+h[m] ·n −

+ , (4.29)

and since

h[m] ·n=−∇um ·n=∇(S∂Ω[−m ·n] +NΩ[divm]) ·n (4.30)

has a jump only in the simple layer potential term, we get (cfr. Proposition 3.24):

h[m] ·n −+ =m ·n and b[m] ·n −

+ = 0. (4.31)

Hence b[m] · n is continuous when crossing ∂Ω (in the normal direction), while h[m] · n has a jump

discontinuity equal to m ·n. The previous two relations (4.31) are known in the physical literature (but

also in the mathematical one) as the transmission or (jump) conditions for the demagnetizing field

h and for the magnetic flux density field b.

4.3 The L2 theory of the demagnetizing field.

In what follow we denote by L2(Ω,R3) the Hilbert space of real square-summable in Ω vector functions

with norm ‖·‖Ω and inner product (·|·)Ω. We also denote by Cχ∞(Ω,R3) the normed subspace of L2(Ω,R3)

made by the zero extensions (outside of Ω) of the C∞(Ω,R3) vector valued functions. From a pure set

theoretical point of view:

Cχ∞(Ω,R3)≡m:R3→R

3 | m∈C∞(Ω),m≡ 0 in Ω′ :=R3\Ω. (4.32)

We now prove the following:

Proposition 4.6. Let Ω be a bounded and smooth domain of R3. The following properties hold:

P1. If m ∈ Cχ∞(Ω, R3) then the magnetic flux density b[m] and the magnetic field h[m] are both

square integrable in all the space: b[m],h[m]∈L2(R3,R3).

P2. The magnetic field is a negatively semi-defined operator with respect to the L2(Ω, R3) inner

product:

−(h[m],m)Ω > 0 ∀m∈Cχ∞(Ω,R3) . (4.33)

P3. Moreover h is a self-adjoint operator with respect to the L2 scalar product, and:

−(h[m], p)Ω =−(m,h[p])Ω = (h[m],h[p])R3 (4.34)

for every m, p∈Cχ∞(Ω,R3).

P4. The fields b[p] and h[m] are orthogonal with respect to the L2 scalar product, id est

(b[p],h[m])R3 =0 ∀m, p∈Cχ∞(Ω,R3). (4.35)


56

Proof. (P1)From the regularity properties (up to the boundary) of the surface potentials we know that

b[m] and h[m] are bounded in Ω and in Ω′. Moreover since the Newtonian potential NΩ[m] is harmonic

in Ω′ we know (cfr. Theorem 3.18) that |∂ijNΩ[m](x)| ∈ O∞(|x|−3). Thus h[m] = −∇divNΩ[m] is in

L2(R3,R3) as well as b[m] = curlcurlNΩ[m].

(P2, P3, P4) We now need to auxiliary results: the first one is to observe that since div b[m] = 0 in

Ω∪Ω′ and b[m] ·n, um are continuous when crossing ∂Ω (due to (4.31)), we have

(b[m],h[m])Ω ≡ −(b[m] ,∇um)Ω = −∫

Ω

div (um b[m])dµ

=−∫

∂Ω

(um b[m]) ·n dσ =

∫

∂Ω′

(um b[m]) ·ndσ

= (b[m] ,∇um)Ω′≡−(b[m] ,h[m])Ω′ .

(4.36)

Therefore (b[m],h[m])R3 = 0 for every m∈Cχ∞(Ω,R3). The second auxiliary result that we need easily

follows from the first one: indeed from the previous orthogonality condition we get that

−(h[m],m)R3 =−(h[m],m− b[m])R3. (4.37)

Hence, since m− b[m] =−h[m],

−(h[m],m)Ω =−(h[m],m)R3 =−(h[m],m− b[m])R3 = ‖h[m]‖R32 . (4.38)

(P2) It is an obvious consequence of (4.38).

(P3)Thanks to the parallelogram law and the linearity of h, we have

‖h[m+ p]‖R32 + ‖h[m− p]‖R3

2 =2(‖h[m]‖R32 + ‖h[p]‖R3

2 ). (4.39)

Taking into account (4.38) we so finish with

(h[m], p)Ω =(m,h[p])Ω ∀m, p∈Cχ∞(Ω,R3). (4.40)

And the previous relation proves the self-adjointness of h. Similarly from the reverse parallelogram law

‖h[m+ p]‖R32 −‖h[m− p]‖R3

2 =4(h[m],h[p]), taking into account (4.38) and (4.40) we get

(p,h[m])Ω =−(h[m],h[p])R3 ∀m, p∈Cχ∞(Ω,R3). (4.41)

This concludes the proof of (P3).

(P4) Finally from (4.41)

(b[p],h[m])R3 = (p+h[p],h[m])R3 = 0, (4.42)


As a direct consequence of Proposition 4.6 it is now possible to extend (by continuity) the domain of

the operator h from Cχ∞(Ω,R3) to L2(Ω,R3). Indeed the following (now trivial) result holds:


57

Theorem 4.7. The operator −h: Cχ∞(Ω, R3) → L2(R3, R3) is bounded when Cχ

∞(Ω, R3) is endowed

with the L2(Ω,R3) norm, and

‖h‖ := sup‖m‖Ω=1

‖−h[m]‖R3 = 1 (4.43)

There exists a unique extension of −h from Cχ∞(Ω, R3) to L2(Ω, R3), and this extension is still self-

adjoint, positively semi-defined, with

inf‖m‖Ω=1

(−h[m],m)= 0. (4.44)

Proof. From the orthogonality condition (4.35) concerning b and h it follows that −h is bounded. Indeed

‖h[m]‖R32 = ‖χΩm‖R3

2 −‖b[m] ‖R32 6 ‖m‖Ω

2 and therefore:

‖−h‖= sup‖m‖Ω=1

‖−h[m]‖R3 6 sup‖m‖Ω=1

‖m‖R3 = 1. (4.45)

Moreover if m1∈Cχ∞(Ω,R3) is conservative in Ω, i.e. irrotational (or curl free) (curlm1 =0) and

normal to the boundary (m1×n|Γ=0), then (4.27) gives−h[m1]=m1 and therefore the supremum in

(4.45) is reached wheneverm1. This proves that ‖−h‖=1. Extension of −h by continuity onto L2(Ω,R3)

results in ‖−h‖=1.

By standard density arguments relations (4.35) and (4.34) are still verified for the extended operator.

Thus the operator −h defined in L2(Ω,R3) and with values in L2(R3,R3) is still self-adjoint and positively

semi-defined.

To prove the last statement we note that due to (4.33) we have (−h[m],m) > 0. Moreover by the

integral representation (4.27), we get that (−h[m0], m0) = 0 whenever m0 is solenoidal in Ω, i.e.

indivergent (or divergence free) (divm0 =0) and tangent to the boundary (m0 ·n|Γ =0).

Remark 4.2. The same extension result can be obtained via the Calderon-Zygmund theory of singular

integrals [Fri80]. Indeed from (4.18), i.e. from the relation

divNΩ[m] =1

4π

∫

Ω

m(y) · ∇y

[

1

|x− y |

]

dτy (4.46)

and the Calderon-Zygmund Lemma (cfr. [Ste71]) we know that there exists a unique bounded operator u

from L2(Ω,R3) into the Sobolev space H1(Ω,R3) which coincides with divNΩ[m] when m∈Cχ∞(Ω,R3).

Moreover for m∈L2(Ω,R3) the weak derivative of um is given by

−h[m] = ∇um =−1

3m+

1

4π

∫

Ω

[

− m(y)

|x− y |3 +m(y) · (x− y)

|x− y |5 (x− y)

]

dτy (4.47)

where the latter integral must be understood in the Cauchy sense: that is as

1

4πlimε→0

∫

Ω\B(x,ε)

[

− m(y)

|x− y |3 +m(y) · (x− y)

|x− y |5 (x− y)

]

dτy (4.48)


58

where B(x, ε) is a ball of radius ε with center at x. Since the kernel of the latter integral is even, −h is

self-adjoint (cfr. [Ste71]).

Remark 4.3. The established properties of −h have an interesting physical interpretation [Fri80]: The

inequality 0 6 −(h[m], m) indicates that the mean angle between the induced field h[m] and the

magnetization m is not less than π/2.

The relation −(h[m] ,m/‖m‖Ω)6 ‖m‖Ω indicates that the mean value of the projection of h along

m is not more than the mean value of m in Ω.

These two relations give a rigorous mathematical interpretation to the well-known maxims among

electrical engineers that «the induced field is directed opposite to the net field, or magnetization», and

the «induced field is less than the magnetization» [Fri80].


59

5The Demagnetizing Factors

In this Chapter a modern and simple proof of the homogeneous ellipsoid problem relative to the Newto-

nian potential is given. The argument is essentially based on the use of coarea formula which permits to

reduce the problem to the solution of the evolutionary eikonal equation , revealing in that way, the pure

geometric nature of the problem. Due to its physical relevance, particular attention is paid to the three-

dimensional case, and in particular to the computation of the demagnetizing factors which are one of the

most important quantities of ferromagnetism.

5.1 Introduction

The computation of the gravitational potential generated by an homogeneous ellipsoid (the homogeneous

ellipsoid problem), was one of the most important problems in mathematics for more than two centuries

after Sir Isaac Newton enunciated the universal law of gravitation in his Philosophiae Naturalis Principia

Mathematica [Bel80, Dan11a, Dan11b, New87, Sha91]. In modern and more general mathematical terms,

fixed a bounded domain Ω of RN, with N >3, the problem consists in finding an explicit expression of the

Newtonian potential generated by a uniform charge/mass density on Ω, given for every x∈RN by [DiB10]

NΩ[χΩ](x) := cN

∫

Ω

1

|x− y |N−2dy , cN :=

1

(N − 2)ωN. (5.1)

where ωN is the surface measure of the unit sphere in RN.

In 1687, Newton showed what nowadays is known as Newton’s shell theorem [New87]: if Ω is an

homogeneous ellipsoid centered at the origin, then for all t> 1, NtΩ\Ω is constant in Ω, i.e. tΩ\Ω (the so

called ellipsoidal homoeoid) induces no gravitational force inside Ω. The shell theorem was an important

step toward the problem of the computation of the gravitational potential induced by an homogeneous

ellipsoid, which was for the first time solved by Gauss in 1813 by the means of what it is in present days

known as the Gauss’s law for gravity [Gau13]. Later (but independently), in 1839, Dirichlet proposed

a solution of the problem based on the theory of Fourier’s integrals [Kro69]. The results of Gauss and

Dirichlet can be summarized by saying that if Ω is an ellipsoidal region centered at the origin, then

NΩ[χΩ](x)= c−Px ·x ∀x∈Ω, (5.2)

61

for some constant c ∈R and some matrix P ∈RN×N whose values can be expressed in terms of elliptic

integrals [Kel10, Str07].

The converse statement (the inverse homogeneous ellipsoid problem) is also true [DF86, DiB10,

Kar94], namely: if Ω is a bounded domain of RN such that R

N\Ω is connected and (5.2) holds, then

Ω is an ellipsoid. Historically speaking the inverse homogeneous ellipsoid problem was for the first

time solved by Dive [Div31] in 1931 for N = 3 and in 1932 by Hölder [Höl32] for N = 2. A modern

proof of this result can be found in DiBenedetto and Friedman [DF86] who, in 1985, extended it to

all N > 2. In 1994, Karp [Kar94], by the means of certain topological methods, obtained an alter-

native proof of the inverse homogeneous ellipsoid problem.

The story of the homogeneous ellipsoid problem is slightly bit different: even though relative modern

treatments can be found in Kellogg [Kel10] and Stratton [Str07], they are both based on the use of

ellipsoidal coordinates (and the separation of variables method) which tend to focus the attention on the

technical details of the question rather than on its geometric counterpart. Aim of this chapter is to give

a modern and simple proof of the homogeneous ellipsoid problem.

More precisely, in section 5.2 we give a new argument for the homogeneous ellipsoid problem based on

the use of coarea formula [EG91] and the notion of an eikonal cover of the space. The approach deserves

his own interest since it reduce the problem to the solution of the evolutionary eikonal equation [AA03,

DiB10, LVS87], and this leads on the one hand (and at least in principle) to treat geometries which are not

confined to the ellipsoidal one, and on the other hand to reveal the pure geometric nature of the problem.

In section 5.3 we apply it to the case of ellipsoidal geometry, showing that indeed (5.2) holds. An

expression in terms of the elliptic integrals is given for the values of the coefficients c and P . Due to

its physical relevance, particular attention is paid to the eigenvalues of P in the three-dimensional

setting. Indeed, when N = 3, the matrix P and its eigenvalues, known in the theory of ferromagnetism

respectively as the demagnetization tensor and the demagnetizing factors , are one of the most important

and well-studied quantities of ferromagnetism [BD03, Bro62b, Osb45]. In fact, the following magnetostatic

counterpart of the homogeneous ellipsoid problem holds: given a uniformly magnetized ellipsoid, the

induced magnetic field is also uniform inside the ellipsoid . This result was for the first time showed by

Poisson [Poi25] while an explicit expression for the demagnetizing factors was obtained for the first time by

Maxwell [Max73]. Their importance is in that they encapsulate the self-interaction of magnetized bodies:

their knowledge is equivalent to the one of the corresponding demagnetization (stray) fields [BD03].

5.2 Main result.

In this section we state and prove the main result of the paper. To this end we start by giving the following

Definition 5.1. Let Ω and ΩT be two C2 domains of RN. A family of C2 domains (Ωt)t∈[0,T ), such that

ΩT =∪t∈[0,T )Ωt, is called an eikonal cover of ΩT starting at Ω, if for every t∈ [0, T ) there exists a family of

C2 functions (ut:Ωt→R+)t∈[0,T ) and a family of associated positive diffeomorphisms (φt:Ω→Ωt)t∈[0,T ),

differentiable with respect to t, satisfying the following conditions:

E1. Inside-outside property. For every [0, T ), the following relations hold

Ωt= x∈RN : ut(x)< 1 and ∂Ωt= x∈R

N : ut(x)= 1. (5.3)

The Demagnetizing Factors

62

E2. Compatibility condition. For every t∈ [0, T ), φ0 = idΩ and φt(Ω)= Ωt.

E3. Eikonal condition. There exists a family of scalar function (vt: R+ →R

+)t∈[0,T ) such that for

every t∈ [0, T ) and every x∈Ωt

vt(x)=−vt(ut(x))∇ut(x) , (5.4)

where we have denoted by vt the so-called velocity field, i.e. the vector field defining the dynamical system

∂tφt=vt φt, φ0 = idΩ, having φt as flux.

Remark 5.1. The attribute eikonal given to such a covering of ΩT is due to the following reason:

whenever ut satisfy the compatibility condition E2 one has ut(φt(x)) = 1 for all x ∈ ∂Ω; differentiating

both members with respect to t we get

∇ut(φt(x)) ·vt(φt(x))+ ∂tut(φt(x)) =0 ∀x∈ ∂Ω, ∀t∈ [0, T ). (5.5)

Applying the eikonal condition (5.4) we get for all x∈ ∂Ω and all t∈ [0, T )

∂tut(x)= vt(ut(x))|∇ut(x)|2, (5.6)

and the previous equation, belongs to the class of evolutionary eikonal equations [AA03, DiB10, LVS87].

In what follows, for every x ∈ RN, we denote by NΩ[ρ](x) the Newtonian potential generated by the

density ρ on Ω:

NΩ[ρ](x) := cN

∫

Ω

ρ(y)

|x− y |N−2dy , cN :=

1

(N − 2)ωN. (5.7)

Theorem 5.2. Let Ω be a bounded C2 domain, and let the family of bounded and C2 domains

(Ωt)t∈[0,T ) be an eikonal cover of ΩT starting at Ω. Then, if div vt= ρt ∀x∈ΩT , for some function ρt:

[0, T )→R, then

a(T )NΩT[1](x) = NΩ[1](x)+

∫

τ(x)

T

a(t)

(

∫

ut(x)

1

vt(s) ds

)

dt , a(t) := exp

(

−∫

0

t

ρs ds

)

(5.8)

where for every x∈ΩT , we have denoted by τ (x) the minimum τ (x)>0 such that uτ(x)(x)61, i.e. such

that x∈Ωt. In particular, τ (x)= 0 for every x∈Ω.

Proof. By Reynold’s transport theorem we get [Sch07]

∂tNΩt[1] = NΩt

[div vt]−∫

Ωt

∇y[E(x− y)] ·vt(y) dτy , . (5.9)

Since Ωt=∪s=01 ut

−1(s), the use of coarea formula (see [EG91]) leads to

−∫

Ωt

∇y[E(x− y)] ·vt(y) dτy = −∫

0

1(∫

∂Ωs

∇y[E(x− y)] · vt(y)

|∇ut(y)|dσy

)

ds. (5.10)

5.2 Main result.

63

The previous relation, together with the eikonal condition given by (5.4), permits to rewrite equation

(5.9) as

∂tNΩt[1] = NΩt

[div vt] +

∫

0

1

vt(s)D∂Ωs[1] ds , (5.11)

where, for every C2 domain G of RN and for every x∈RN\∂G we have denoted by D∂G[1](x) the classical

double layer potential generated in x by a distribution of dipoles identically equal to 1 on ∂Ω.

D∂G[1](x) :=− 1

ωN

∫

∂G

x− y

|x− y |N ·n(y) dσ(y) (5.12)

If in addition div vt is uniform in space (i.e. if div vt(x) = ρt for some ρt: [0, T )→R and every x ∈ΩT),

we have

∂tNΩt[1]− ρtNΩt

[1] =

∫

0

1

vt(s)D∂Ωs[1] ds. (5.13)

Thus, setting

a(t) := exp

(

−∫

0

t

ρs ds

)

(5.14)

we get

∂t(a(t)NΩt[1])= a(t)

∫

0

1

vt(s)D∂Ωs[1] ds (5.15)

We now recall that due to a well-known consequence of the Green’s representation formula, also known in

the physical literature as Gauss’s law for gravity [DiB10], we haveD∂G[1](x)=χG(x) for every x∈RN\∂G.

Therefore, coming back to the case ∂Ωs = ut−1(s), we have D∂Ωs

[1](x) = χ[ut(x),1)(s) · χ[τ(x),∞)(t) for

almost every x∈RN and every t>0, where we have denoted by τ(x) the minimum of the τ (x)∈R

+ such

that uτ(x)(x) 6 1. In particular one has τ (x) = 0 for every x ∈ Ω. Substituting the expression found for

D∂Ωs[1](x) into (5.11), we get (5.8), i.e. the first claim of the theorem. Thus, integrating both members

of (5.15) between zero and T , we get (5.8).

5.3 The Homogeneous Ellipsoid Problem and the Demagnetizing Factors.

In this section we apply Theorem 5.2 to the case in which Ω is an N -dimensional ellipsoidal domain of

semi-axes lengths a1, a2, ..., aN ∈R+, i.e.

Ω := x∈RN : u(x)6 1 , u(x) := ξx ·x , ξ :=diag

(

1

a12 ,

1

a22 , ...,

1

aN2

)

∈RN×N. (5.16)

To this end we look for an eikonal cover of RN starting at Ω. We start by looking for a family of functions

of the form ut(x) = ξtx · x, with ξt= diag[a1(t), a2(t), ..., aN(t)], satisfying the eikonal equation (5.6), i.e.

such that ξt′x ·x=4vt(ξtx ·x)ξt2x ·x. Since a degree of freedom is on vt, we choose vt(ξtx ·x)=−1/4, and

we look for a decoupled solution of the previous problem, i.e. a solution the Cauchy problem ξt′ = −ξt2,

ξ0 = ξ, which, for every t> 0, is given by ξt= (ξ−1 + tI)−1. Therefore for every t> 0 and every x∈RN

ut(x)=x1

2

a12 + t

+x2

2

a22 + t

+ ···+ xN2

aN2 + t

. (5.17)


64

Since ρt :=div vt=1

4∆ut=

1

2

∑

i=1N 1

ai2 + t

, we have

a(t)= exp

(

−1

2

∑

i=1

N ∫

0

t 1

ai2 + s

ds

)

=∏

i=1

Nai

(ai2 + t)1/2

.

Thus, once observed that for every x ∈ RN we have limT→+∞ a(T )NΩT

(x) = 0, we reach the following

expression for the Newtonian potential generated by the general ellipsoid Ω:

NΩ(x)=1

4

∫

τ(x)

+∞

[1−ut(x)]∏

i=1

Nai

(ai2 + t)1/2

dt (5.18)

The integral (5.18) can be evaluated by the means of the theory of elliptic integrals.

We know focus on the three-dimensional framework (N = 3) and in particular: on the expression

assumed by the Newtonian potential in the internal points of Ω, and on the demagnetizing factors of

the general ellipsoid. To this end we recall that the stray field hd[m] associated to a magnetization

m belonging to the Sobolev space Hdiv1 (Ω,R3) can be expressed as the gradient field associated to the

magnetostatic potential ϕm(x), i.e for every x∈R3

ϕm(x) :=1

4π

∫

Ω

(divm)(y)

|x− y | dy− 1

4π

∫

∂Ω

m(y) ·n(y)

|x− y | dσ(y) , hd[m] :=−∇ϕm. (5.19)

In particular, if m is constant in Ω, the use of divergence theorem leads to the following expression of

ϕm in terms of the Newtonian potential: ϕm(x)=−m(x) · ∇NΩ(x). Therefore from (5.18)

ϕm(x) =Px ·m , hd[m] =−Pm ∀x∈Ω, (5.20)

where we have denote by P the diagonal matrix, known in literature as demagnetizing tensor , whose

diagonal i-entry (the i-th demagnetizing factor) is given for every i∈N3 by

Pi :=1

2

∫

0

+∞ 1

(ai2 + t)

∏

j=1

3aj

(aj2 + t)1/2

dt. (5.21)

Obviously Pi>0 for every i∈N3 and if a1>a2>a3 then P1 6P26P3. The trace of P satisfy the relation

tr(P ) = 1: this can be verified by a direct evaluation of the integrals (5.21), or by simply observing that

since the Newtonian potential UΩ satisfies the Poisson equation ∆NΩ=−χΩ in the sense of distribution on

Ω, one has trP=div (Px)=−∆NΩ=χΩ. Assuming a1>a2>a3, from the theory of elliptic integrals we get

P1 = 1−P2−P3 (5.22)

P2 = − a3

a22− a3

2

[

a3− a1a2

(a12− a2

2)1/2E

(

arccos

(

a2

a1

)∣

∣

∣

∣

a12− a3

2

a12− a2

2

)

]

(5.23)

P3 = +a2

a22− a3

2

[

a2− a1a3

(a22− a3

2)1/2E

(

arccos

(

a3

a1

)∣

∣

∣

∣

a12− a2

2

a12− a3

2

)

]

, (5.24)

where, for every φ∈R and every 0< p< 1 we have denoted by

E(z |p) :=

∫

0

z

(1− p sin2θ)1/2 dθ (5.25)

5.3 The Homogeneous Ellipsoid Problem and the Demagnetizing Factors.

65

the incomplete elliptic integral of the second kind expressed in parameter form . In particular, in the case

of a prolate spheroid (a1 > a2 = a3) we get

P1 =− a32

(a12− a3

2)3/2

[

(a12− a3

2)1/2 + a1 arccoth

(

a1

(a12− a3

2)1/2

)

]

, P2 =P3 =1−P1

2(5.26)

while in the case of an oblate spheroid (a1 = a2 > a3)

P1 =P2 =1−P3

2, P3 =

a12

(a12− a3

2)3/2

[

(a12− a3

2)1/2 + a3 arctan

(

a3

(a12− a3

2)1/2

)

− a3π

2

]

. (5.27)

Finally, in the case of a sphere (a1 = a2 = a3) one finish with P1 =P2 =P3 =1

3.


66

6Micromagnetics

«A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates,

and the more extended is its area of applicability. Therefore the deep impression which classical thermodynamics madeupon me. It is the only physical theory of universal content concerning which I am convinced that within the frameworkof the applicability of its basic concepts, it will never be overthrown.»

Albert Einstein

Our introduction leans heavily on two seminal works of William Fuller Brown titled Micromagnetics

[Bro63] and Magnetostatic Principles in Ferromagnetism [Bro62b]. This is quite inevitable as we agree

completely on the statement of the problem.

6.1 The general problem

A ferromagnetic material may be defined as one that possesses a spontaneous magnetization: that is,

sufficiently small volumes of it have an intensity of magnetization (magnetic moment per unit volume)

Ms(T ) dependent on the temperature but independent, or at least only slightly dependent on the presence

or absence of an applied magnetic field6.1. The existence of this spontaneous magnetization is explained

by the Weiss molecular field postulate, amended quantum-mechanically by Heisenberg; the amendment

replaces the mysterious molecular field by exchange forces, which are less mysterious or more according

to one’s feeling toward quantum mechanics. But this theory , based on exchange forces that tend to align

the spins and thermal agitation that tends to misalign them, says nothing about the direction of the

vector magnetization M; only that its magnitude must be Ms(T ).

Experimentally, it is observed that though the magnitude of |M| =Ms(T ) is uniform throughout a

homogeneous specimen at uniform temperature T , the direction of M is in general not uniform, but varies

from one region to another, on a scale corresponding to visual observations with a microscope. Uniform

of direction is attained only by applying a field, or by choosing as a specimen, a body which is itself of

microscopic dimensions (a fine particle); the evidence of uniformity in the latter case is indirect but

convincing (see Chapters 8 and 9).

6.1. By applied magnetic field we shall always mean the field of magnetizing coils or magnets (or both) externalto the specimen, as distinguished from the field (be it the H field or the B field) produced by the magnetization of thespecimen under consideration.

67

The tendency of a ferromagnetic specimen to break up into domains, with their vector magneti-

zation oriented differently in any such domain, explains the possibility of a demagnetized state; and in

fact such a domain structure was postulated by Weiss in order to reconcile his theoretically predicted

spontaneous magnetization with the experimental possibility of demagnetization. Today the evidence

of domain structure are so many and so inescapable that its status is no longer that of a postulate, but

rather of an experimental fact (see Figure 1.1).

In two respects, however, the range of validity of this fact has at times been supposed more universal

than it actually is. First, domains were for a long time tacitly assumed to be present in all specimens,

regardless of their geometry. This naive assumption delayed the theoretical understanding and practical

application of the properties of fine particles. Second, domains have often been discussed as if they were a

phenomenon to be expected in all ferromagnetic materials. Actually, both theory and experiment indicate

that domains in the usual sense – regions within which the direction of the spontaneous magnetization

is uniform or at least nearly so – do not occur unless there are present strong «anisotropy» forces, which

cause certain special directions of magnetization to be preferred. When such forces are absent or weak, the

magnetization direction, over dimensions comparable with the usual domain dimensions, varies gradually

and smooth. It is therefore clear that domain structure, though normal, is not universal.

More generally, once defined m := M/|M| = Ms−1M, we should suppose merely that the direction

cosines m1,m2,m3 (subject to the constraint m12+m2

2+m32=1), are functions of the point x at which the

vector magnetization M is being evaluated. Whether m is constant or variable, in the region occupied

by the ferromagnetic body, continuous or discontinuous, step-functions or sinusoids, need not be decided

until later.

The general problem to be examined in Micromagnetic theory is the problem of developing a theory

of this magnetic microstructure, concerning which the Weiss-Heisenberg theory is noncommittal. Ideally,

one would hope by such methods to make present domain theory, with its incomplete arguments and often

circular reasoning, obsolete. Domains, when they exist, should in principle emerge from the theory without

having to be postulated. In this respect, Micromagnetics has contributed some new concepts that seem

likely to prove valuable and it can provide precisely formulated methods of approximation and precise

criteria for assessing the approximations of domain theory. Quite possibly the value of the Micromagnetics

approach lies as much in these contributions as in its mathematical solutions of specific problems [Bro63].

6.1.1 Forces involved

The most basic method of solving this problem would be to use an atomic model, such as a lattice of spins,

and to introduce into the model those forces that the Weiss-Heisenberg theory left out. As has already

been mentioned, that theory takes account only of exchange forces and thermal agitation. The known

forces that remain to be introduced are: magnetic dipole-dipole forces; forces due to spin-orbit

coupling and to magnetic quadrupole and higher moments; modifications of the exchange forces that

result when the directions of neighboring spins are not exactly parallel; and magnetostrictive forces,

which are not physically distinct from the ones already enumerated but are the modifications of them

that come about because of the ability of the lattice to undergo strains. Available methods of treating

ferromagnetism atomically are already inadequate when only exchange forces and thermal agitation are

taken into account; they become quite unmanageable when, for example, magnetic dipole-dipole forces

are introduced. Accordingly, we must resort to a phenomenological type of theory .

Micromagnetics

68

The possibility of such a theory rests on the fact that all these new forces have only a small perturbing

effect on the parallelism (or, in certain cases, anti-parallelism) of neighboring spins. The spin direction,

in other words, can change only by a small angle from one lattice point to the next. It therefore seems

legitimate to approximate the direction angles of the spins (or more conveniently of the associated

magnetic moments) with continuous functions of position. By this device, exactly analogous to the

replacement of individual atomic masses by a continuous density in elementary mechanics, sums over

lattice points are replaced by integrals over a volume, and the techniques of calculus become applicable.

The basic concept in such a theory is a vector magnetization M =Ms(T )m whose direction cosines

(expressed by m) vary continuously with position. Changes through appreciable angles may occur on a

scale that is small in comparison with the domain scale, or on a scale that is comparable with the domain

scale; these two cases will correspond, respectively, to the case in which domains in the ordinary sense

are observed and to the case in which only a gradual variation is observed.

The detailed development of such a theory is the theme of Micromagnetics. The theory is far from

being fully developed; all that can be said is that the foundations have been laid.

6.1.2 The variational approach

In the foregoing discussion, the term «force» has been used in a general sense. If magnetostriction is

ignored, our model is a rigid specimen occupying a region of space Ω, with a vector magnetization M:

Ω→R3, defined in Ω, whose direction varies continuously with the point x∈Ω, but whose magnitude has

a value Ms(T ) determined by the temperature only. The «forces» are then torques (couples) that act on

the magnetization vector distribution M. In thermodynamic equilibrium, the orientation of M(x) at each

point x∈Ω must be such that the total torque on each moment M(x) is zero. When the field is changed,

the torques in the old orientations usually cease to be zero; then the dissipative processes that tend toward

thermodynamic equilibrium will ultimately establish a new equilibrium distribution of orientations.

Though direct use of torques is sometimes convenient, energy methods are generally more powerful.

To this end, we shall set the problem in the context of classical thermodynamics, where some universal

variational relation concerning the free energy, is (physically) assumed to hold. By free energy , as

distinguished from energy , we mean the Helmholtz free energy F =U −ST or some other thermodynamic

potential in which the natural independent thermal variable is T , as distinguished from U or some

other thermodynamic potential in which the natural thermal variable is S. Indeed, in the description of

ferromagnetic phenomena the convenient independent thermal variable is absolute temperature T rather

than entropy S.

The thermodynamic (variational) approach leads the problem to the searching for analytic expressions

of the energy terms that contribute to the free energy. Our methods of arriving at such expressions will

be partly microscopic and partly phenomenological, but the energy expressions, once obtained,

will be regarded entirely phenomenologically. If this theory were to be developed on the basis of a truly

atomic model, a conceivable procedure would be to evaluate the partition function, and from it the free

energy, by the methods of statistical mechanics. Though formal expressions in the form of sums over

states and lattice sites might be derived without too much trouble, the reduction of these to usable forms

is out of the question. Furthermore, even if this procedure could be carried out, it would not, at least

without basic modification, lead to a theory of magnetic hysteresis; for standard statistical mechanics

yields only states of complete thermostatic equilibrium, and magnetic remanence is not such a state.

6.1 The general problem

69

In Section 6.1.1 we classified the forces according to their physical origin. In a phenomenological

theory, it is more convenient to classify them according to the mathematical form of the free-energy

expressions that describe them. In a rigid cubic crystal, the dipole-dipole forces correspond to free-

energy expressions similar in form to the energy integrals of formal magnetostatic theory; spin-orbit

and quadrupole forces, to free-energy densities dependent on the local direction of magnetization; and

the exchange forces, as perturbed by non-uniformity of magnetization, to free-energy densities depen-

dent on the spatial gradients of the direction cosines (or direction angles) of the magnetization. These

three contributions to the free energy are usually called, in order, the magnetic or magnetostatic

energy, the anisotropy or crystalline-anisotropy or magneto-crystalline anisotropy energy,

and the exchange or exchange-stiffness energy. The terminology is poor, for it confuses classification

according to origin and classification according to form; but in cubic crystals it causes little trouble. In

hexagonal crystals, on the other hand, the dipole-dipole forces contribute, besides the formal magneto-

static-energy integral, a term in the form of a free-energy density dependent on the local magnetization

direction. In a phenomenological theory, concerned with forms and not with origins, this term must be

treated as part of the «anisotropy energy» and in fact cannot be distinguished from other terms of the

same form but of different origin; thus an energy term of magnetic origin is included in the «anisotropy

energy» and not in the «magnetic energy». We shall not try to reform the established terminology; no

confusion will occur if we remember that our theory is phenomenological and that our classification is on

the basis of form, not of origin.

6.2 Thermodynamic relations.

We begin with a brief summary of the principal thermodynamic relations valid for a generic system with

homogeneous properties. Later, we will specialize this description to the magnetic case. In doing this, we

will follow a variant of the Gibbs axiomatic approach, as exposed in [GS70], [Cal85] and [Tsc00]. Here the

concept of internal energy is taken for granted and some of its properties are assumed in an axiomatic

way, so that attention is turned on deducing rigorous consequences therefrom.

6.2.1 The internal energy state function.

Every macroscopic physical system is characterized by particular states, known as thermodynamic

equilibrium states, in which an closed system remains unchanged until some condition is changed

inside or outside of the system. We assume that the complete state space of a thermodynamics system T(which includes also the non-equilibrium states) can be described by the product vector spaceR×W×Q,

in which G :=R×W represents the set of possible equilibrium states of T , also known in literature as

the Gibbs state space.

Thus, the state space of T can be characterized by the values of three quantities: the entropy S ∈R,

the vector of work variables x∈W and the vector of uncompensated heats q. In particular, the

set of equilibrium states G, can be characterized by the entropy S ∈R and the vector of work variables x.

The thermodynamic properties of T can then be deduced from the internal energy state function

U defined on G × X (and with values in R), through the functional relation U = U(S, x, q), whose

restriction to G, denoted by u=u(s,x), satisfies the following relations:

T1. u(λs, λx)=λu(s,x) for every λ> 0 (1-homogeneity)

T2. u(s1,x)>u(s2,x) if s1>s2 (strict monotonicity)

T3. u((s1,x1) + (s2,x2))6 u(s1,x1) +u(s2,x2) (subadditivity)

Micromagnetics

70

The 1-homogeneity, is a consequence of the postulate of extensivity of internal energy u: geometrically it

means that u is a ruled surface. The strict monotonicity property is related to positivity of temperature,

which we define as T := ∂su. The subadditivity expresses physically the decreasing property of energy

in isolated systems when internal constraints are released; equivalently: for an isolated system of fixed

u and x, the state of unconstrained equilibrium has an entropy greater than all corresponding states of

constrained equilibrium. The importance of the subadditivity relation relies on the following immediate

consequence: by subadditivity and 1-homogeneity, with λ=1

2, one has

u( s1 + s2

2,

x1 + x2

2

)

61

2u(s1,x1) +

1

2u(s2,x2) , (6.1)

i.e. u is a convex function. By well known theorem on convex functions it follows that u is continuous if

u is measurable; right and left first partial derivatives of u always exist, and the corresponding differen-

tiated variables are monotonically decreasing (so that they may have at most jump discontinuities in a

enumerable set).

6.2.2 The First law of Thermodynamics.

In what follows we assume that U ∈C1(G×Q). Under this hypothesis, for every smooth thermodynamic

process Γ: t ∈ [tA, tB] 7→ (st, xt, qt) starting in the equilibrium state (sA, xA) and having for final

equilibrium state (sB,xB), letting U(t)=U(st,xt, qt), we have:

u(sB,xB)− u(sA,xA) =

∫

tA

tB

U (t) dt =

∫

tA

tB

∂sU · st+∇qU · q t + ∇xU · xt dt (6.2)

We rewrite the previous relation in the more natural form

∆u=

∫

Γ

TdS+ δQ′−h · dx =

∫

Γ

δQ+ δQ′+ δW , ∆u := u(sB,xB)− u(sA,xA), (6.3)

where, we have denoted by δQ :=TdS the generated heat (with T := ∂sU the generalized absolute

temperature), by δQ′ the internally generated heat, and by δW :=−h · dx the work performed

on the system (h :=−∇xU being the x-conjugate state variable which, from a physical point of view,

characterizes the external actions exerted on the system).

In particular, for quasistatic processes, i.e. for processes Γ such that Γ(t) belongs to the Gibbs state

space G=R×W for all t∈ [tA, tB], the internally generated heat δQ′ vanishes and the integral relation

(6.3) becomes nothing more that the well known first law of thermodynamics. Indeed, with the

positions δq :=Tds and δw :=−h · dx, the differential form du reads as du= δq+ δw, and its associated

integral relation now reads as

∆u = q(Γ)+w(Γ) =

∫

Γ

δq+ δw. (6.4)

This law expresses the conservation of energy and states that, under a quasistatic transformation Γ where

work δw is performed on the system and heat δq is absorbed by it, the balance equation (6.4) holds.

Next we observe that due to the strict monotonicity condition of u, it is possible to express the entropy

associated to equilibrium states as a functional relation s := s(u,x), and moreover, for every quasi-static

process Γ, the quantity ds is an exact differential, i.e.

sB− sA=

∫

Γ

ds (6.5)


71

for every quasi-static process joining the equilibrium state A= (sA,xA) to B = (sB,xB). This property

is no more true whenever non quasistatic processes are considered.

6.2.3 The second law of thermodynamics: irreversible transformations.

Since so far we have nothing still said about the internally generated heat δQ′: the classical expression of

the first law of thermodynamics does not take into account it. As we will see soon δQ′ is intimately related

to the irreversibility of real physical processes. The concept of irreversibility is rooted in equilibrium

thermodynamics, and the reason that at least some astects of irreversible processes can be treated in

equilibrium thermodynamics is the following: For every irreversible (i.e. real physical) process proceeding

from an equilibrium state A to a new equilibrium state B, a reversible process may be devised which

has the same initial equilibrium state A and final equilibrium state B. Thus, a quasi-static process may

be substituted for the real physical process which is a temporal evolution of both equilibrium and non-

equilibrium states [Tsc00].

Our aim is to show how the δQ′ term is a byproduct of the irreversibility of physical processes. To

this end, we start by noting that for every cyclic process Γ: t∈ [tA, tB] 7→ (St,xt, qt) in which T is constant

and the barrier remains rigid (i.e. for every cyclic process for which xt= 0), one has due to (6.3):

0=∆U

T=

∫

Γ

ds+δQ′

T=

∫

Γ

δQ

T+δQ′

T(6.6)

The differential form δQ′/T is in literature referred to as the internally entropy production associated

to the process Γ.

AB

Γ′

Γqs

Gibbs state space G

Q

WR

Figure 6.1. We consider the cyclic process Γ′∪Γqs made by the union of a generic process Γ′ and a quasi-staticprocess Γqs.

Micromagnetics

72

Next we observe that, for any cyclic process Γ′ ∪ Γqs made by the union of a generic process Γ′ and a

quasi-static process Γqs (see Figure 6.1) we have:

sB− sA =

∫

Γqs

ds =

∫

Γ′

δQ

T+

∫

Γ′

δQ′

T, (6.7)

where in the first equality we have taken into account that the quantity δQ′ is zero in quasistatic processes.

Eventually assuming that in a real physical process (a spontaneous natural process) the entropy δQ′/T

created in the interior of the system is never negative, we have

∫

Γ′

δQ′

T> 0 and therefore sB− sA>

∫

Γ′

δQ

T. (6.8)

What we have reached under some restrictive hypotheses on the nature of the processes, it is actually

assumed (in equilibrium thermodynamics) to have universal validity. Indeed the second law of ther-

modynamics in the so-called Clausius form states that:∮

Γ

δQ

T60 , (6.9)

for every (quasistatic or not) cyclic process Γ [Tsc00].

6.2.4 Thermodynamic potentials for magnetic media.

Let us consider a ferromagnetic body occupying a region Ω ⊆R3, subject to an external magnetic field

H0, and in contact with a thermal bath at constant temperature T (see Figure 6.2). We assume that

around any x ∈ Ω there exists a physically small volume element τ (x) which can be considered as a

thermodynamic system whose equilibrium states are characterized by an internal energy u(x) satisfying

the thermodynamic axioms. For any x∈Ω we assume that u is a function of the entropy s(x) and of the

vector state M(x), which plays the role of the average magnetic moment in τ (x).

Ω

Thermal bath T

x

M(x)

H0

I

Figure 6.2. A ferromagnetic body occupying a region Ω ⊆R3, subject to an external magnetic field H0, and in

contact with a thermal bath at constant temperature T


73

By assuming that µ0H0 and M(x) are conjugate work variables, i.e. that δw= µ0H0 ·dM with µ0H0 :=

∂Mu, according to the first law of thermodynamics (6.4):

∆u(s(x),M(x)) = q(Γ)+w(Γ)=

∫

Γ

δq+ µ0

∫

Γ

H0 · dM , (6.10)

for every quasistatic process Γ. Since u is convex, by a suitable Legendre transform (s↔ T := ∂su) we

can characterize the internal energy u in terms of a function of T and M(x). This leads to the so-called

Helmholtz free energy density at x∈Ω defined by

f(T ,M(x))= u(T ,M(x))−T s(T ,M(x)). (6.11)

Now, we note that given any two equilibrium states A=(T ,MA) and B= (T ,MB) we have

∆f := fB− fA=∆u−T∆s.

Letting Γ be any quasistatic process joining the equilibrium state A to the equilibrium state B, according

the first law of thermodynamics, we have ∆f = q(Γ)+w(Γ)−T∆s. Moreover, due to the second law of

thermodynamics (6.8): q(Γ)6T∆s and therefore

∆f 6w(Γ) = µ0

∫

Γ

H0 · dM . (6.12)

Furthermore, when is Γ is such that µ0H0(t) := ∂Mu(st,Mt) is constant along Γ, then ∆f 6 µ0H0 ·∆M.

This last inequality suggests to consider the potential free-energy density at x∈Ω:

gL(T ,M(x)) := f(T ,M(x))− µ0H0 ·M(x). (6.13)

The density gL is called the Gibbs-Landau free-energy density at x∈Ω, and its importance is due to

the following property: by computing the finite difference ∆gL, we get (by noting that µ0H0 ·∆M=W (Γ))

∆gL =∆f − µ0H0 ·∆M, (6.14)

for every quasistatic process Γ joining A to B, and such that H0(t) is constant along Γ. Therefore, as a

consequence of (6.12), we finish with the fundamental inequality

∆gL6 0. (6.15)

Integrating the previous equations on the region Ω occupied by the ferromagnetic body, we get that for

every thermodynamics process which brings the system from the state MA to the state MB,

GL(T ,MB)6 GL(T ,MA) (6.16)

where

GL(T ,M) :=

∫

Ω

gL(x) dτx = F(T ,M) − µ0

∫

Ω

H0 ·M(x) dτx (6.17)

and

F(T ,M) := U(t,M)−∫

Ω

Ts(T ,M(x)) dτx =

∫

Ω

u(M(x), T )−Ts(T ,M(x)) dτx . (6.18)

Micromagnetics

74

The functional F is referred to as the Helmholtz free energy functional, while GL is referred to as

the Gibbs-Landau free energy functional. The free energy F may be considered as the potential

energy, under isothermal conditions, of a system consisting of the specimen alone; the Gibbs-Landau

free energy functional GL may be considered the potential energy, under isothermal conditions, of

a system consisting of the specimen plus an ideal permanent magnet that produces the field.

Relation (6.16) states that in natural changes, GL can only decrease. Therefore, the states in

which GL is minimum, represent stable equilibrium states. Thus, if we have a formula for GL as a function

of T and M, we can find the stable equilibrium values of the internal coordinates by minimizing GL with

respect to M. Usually the number of minima exceeds one, and therefore the state of magnetization is

not uniquely determined. If |H0| is sufficiently increased, the aligning effect of H0 dominates and reduces

the number of stable equilibrium states to one; on subsequent decrease of H0 to its original value, the

system may find itself in a state different from the initial state. This principle accounts for hysteresis; it

is illustrated most simply by the theory of uni-axial single-domain particles developed by Stoner and

Wohlfarth [SW48]. For specimens capable of a domain structure, we must expect a multitude of stable

equilibrium states. Over sufficiently long time intervals, transitions between them occur; an equilibrium

statistical-mechanical calculation, if it could be carried out, would give the long-time average over the

various states. But the time constants for transition are so long that for most purposes the equilibrium

of any one stable equilibrium state may be regarded as permanent [Bro63].

Our procedure for finding a minimum of GL will be as follows. We seek first an expression for F ,

which may be regarded as the internal potential energy, in the mechanical sense, under isothermal

conditions. Since M = Ms(T )m, with |m| = 1, the internal coordinate that appear in this expression

will be the normalized magnetization m. Our methods of arriving at such expressions will be partly

microscopic and partly phenomenological; but the expressions, once obtained,will be regarded entirely

phenomenologically [Bro63]. Generally speaking, the process of finding a minimum of GL consists of

two steps: finding a state for which the first variation of GL vanishes, and showing that for this state the

second variation is positive, in each case for arbitrary variations of m = (m1, m2, m3), consistent with

|m|2 =1. The first step selects the equilibrium states, and the second tests their stability.

6.3 Free-Energy Formulas

We now require explicit expressions for the various terms in the Helmholtz free energy functional F .

In the attempt to derive such free-energy expressions, there are two possible approaches, which may be

used singly or in conjunction. One approach is to assume, at given temperature, a series in the relevant

variables, e.g. in the direction cosines m1, m2, m3; truncate the series after a few terms, in the hope

that these will prove sufficient; and use crystalline-symmetry considerations to decrease the number of

(temperature-dependent) parameters in the formula. This is the method usually used for evaluating

the anisotropy energy. The other approach is to use an atomic model, perhaps drastically simplified,

to obtain an expression for a particular term in the internal density energy u at T = 0, where thermal

agitation does not complicate the calculation. The expression thus obtained may also be considered an

expression for F at T =0. It may be adapted to arbitrary T by replacing the constants in the formula by

temperature-dependent parameters. This is the method that is convenient for evaluating the contribution

of dipole-dipole forces to the magnetic and anisotropy energies. In either case, the temperature-dependent

parameters in the formula must be evaluated primarily by analysis of experimental data; atomic models,

however, facilitate the estimation of orders of magnitude.


75

6.3.1 The magnetostatic self-energy term

Magnetostatic interactions represent the way the elementary moments interact over «long» distances

within the ferromagnetic specimen. In fact, the magnetostatic field at a given location within the body

depends on the contributions from the whole magnetization vector field M. To evaluate the magnetostatic

self-energy term, we use the microscopic method. For a lattice of dipoles at T =0 the potential energy is

Um :=−µ0

2

∑

x

m(x) ·Hl(x) (6.19)

where Hl is the local field intensity at the position of dipole x due to all the other dipoles (see Section

2.6). We assume that from one dipole to the next the direction of m(x) varies so slowly that over a

physically small sphere6.2, the vectorm may be taken to be either constants or at worst linear functions

of the position coordinates of the dipole i; because of the strong exchange forces that couple one spin

to the next, this should be a good approximation in ferromagnetic materials. Then by the well-known

argument of Lorentz (see Section 2.6)

Hl(x) :=Hd[M](x)+1

3M(x)+hΩS

(x), (6.20)

where Hd[M](x) is the macroscopic field intensity at x computed from the poles due to M, and where

hΩS(x) :=

∑

i∈ΩS\x

hi(x) (6.21)

is the field of the dipoles within a physically small sphere ΩS about dipole x. For a cubic lattice, hΩS=0;

for a crystal lattice in general, hΩS= ΛM, where Λ is a tensor whose form is determined by the crystal

symmetry and whose trace is zero. On substituting (6.20) in (6.19) and replacing sums by integrals, we get

Um(T ,M) :=−µ0

2

∫

M ·(

Hd[M] +1

3M+ ΛM

)

dτ . (6.22)

Here M is the magnetization at the temperature considered, namely (so far) absolute zero, for which

F =U (see equation (6.18)). We now assume that at any temperature T , F contains a term of the form

(6.22), with M the magnetization at temperature T . Then M · M = Ms2(T ), a constant at given T ;

therefore the second term in (6.22) is constant in our variational procedures and may be dropped. Finally,

the integrand M ·ΛM is of the form of a free-energy density dependent only on the local direction of the

magnetization; it may therefore be absorbed into the «anisotropy» term in F . Thus

Fm(T ,M) :=−µ0

2

∫

M ·Hd[M]dτ . (6.23)

We observe that magnetostatic self-energy expresses a non-local interaction. Indeed, the magnetostatic

field functionally depends, through the integral operator Hd (cfr Chapter 4), on the whole magnetization

vector field M, as we anticipated in the beginning of the section.

6.2. A «physically small» sphere is usually defined as one whose radius is large in comparison with the lattice spacing butstill small on the scale of ordinary observations. Here we must replace the second requirement by the more stringent onethat the radius be small on the scale of domain observations. In fact, we should like it to be small in comparison with thethickness of a domain wall and when we cannot satisfy this last condition, we should have some misgivings about the use ofthe Lorentz formula at points inside a wall. We can, however, usually satisfy all these conditions reasonably well by takingthe radius to be about 10 lattice spacings.

Micromagnetics

76

Remark 6.1. The use in (6.23) of the dipolar field Hd[M], rather than the Amperian current field given

by Bd[M] = µ0(Hd[M] +M), is wholly arbitrary. Substitution of Hd[M] = µ0−1Bd−M in (6.23) gives a

constant plus

Fm′ (T ,M)=−1

2

∫

M ·Bd[M] dτ . (6.24)

The choice between these two formulas is one of convenience alone.

6.3.2 The Anisotropy energy term

In most experiments one can generally observe that certain energy-favored directions exist for a given

ferromagnetic specimen, i.e. certain ferromagnetic materials, in zero external field, tend to be magnetized

along precise directions, which in literature are referred to as easy directions.

Physically speaking, the existence of such easy directions is due to the spin-orbit interaction which

acts to couple electron spins to the structure of the crystal lattice, consequently the magnetic moments

are coupled to a certain crystallographic axes and this results in a preferred direction of the magnetic

moments. The easy-axes depend on the atomic structure of the material. Iron for example has a cubic

structure which results in three easy directions coincident with the three crystallographic axes. Whereas

cobalt has an hexagonal lattice structure which generally results in a single easy-axis parallel to the c-

axis of the crystal.

The energy required to rotate a spin system away from the easy direction is called the anisotropy

energy or the magneto-crystalline energy, and is just the energy required to overcome the spin-orbit

coupling. In a phenomenological setting, the anisotropy energy, can be expressed as

Wa(T ,M)=

∫

Ω

wan(m) dτ (6.25)

where m := M/|M| = Ms−1(T )M. The function wan is called the anisotropy energy density and is

assumed to be a non negative even function wan:S2→R

+, defined in the unit spherical surface S2 of R3,

that vanishes only on a finite set of unit vectors, the easy axes. In other terms, the easy axes correspond

to the minima of the anisotropy energy density wan, whereas saddle-points and maxima of wan determine

the medium-hard axes and the hard axes respectively.

uni-axial anisotropy cubic anisotropy

κ1> 0κ1< 0 κ1> 0 κ1< 0

Figure 6.3. (left) Uni-axial anisotropy energy density: if κ1 > 0 the anisotropy energy admits two minima whenthe magnetization lies along the positive or negative u direction with no preferential orientation; if κ1 < 0 anydirection in the plane orthogonal to u corresponds to an easy direction. (right) Cubic anisotropy energy density:if κ1 > 0 there exist six equivalent energy minima corresponding to the directions m∈±u1,±u2,±u3; if κ1 < 0

there exist eight equivalent minima along the directions pointing the vertices of the cube, and the coordinate axesdirections become now hard axes.


77

To evaluate the anisotropy density wan, we use phenomenological methods. We assume a power series

in the directions cosines of m, use the crystal symmetry to decrease the number of coefficients, and

truncate the series after the first two non-constant terms.

The most common anisotropy effect is connected to the existence of one easy direction only, and it

is often called uni-axial anisotropy. In this case the anisotropy free energy density wan is rotationally

symmetric with respect to the easy axis (unit vector) u and depends on the relative orientation of m

with respect to u only. A power series expansion gives for uni-axial crystals, an anisotropy energy density

ϕan(m) =κ0 +κ1|m×u|2 = κ0 + κ1[1− (m ·u)2] . (6.26)

When κ1> 0 the anisotropy energy admits two minima at m=±u, that is when the magnetization lies

along the positive or negative u direction with no preferential orientation. This case is often referred to

as easy axis anisotropy (see the left of Figure 6.3). Conversely, when κ1< 0 the anisotropy density is

minimized form orthogonal to u, meaning that any direction in the plane orthogonal to u corresponds to

an easy direction. For this reason, this case is often referred to as easy plane anisotropy. The constants

κ0 only ensure that the anisotropy energy remains positive in both cases. The constants κ0 and κ1 are

both functions of T .

For cubic crystals three privileged directions exist, and in literature it is referred to as cubic

anisotropy. Denoting by u1, u2, u3 the three easy axes, a power series expansion gives for cubic

crystals an anisotropy energy density

wan(m)= κ0 + κ1

∑

i=1

3

|m×ui|2|m×ui+1|2, u4 :=u1. (6.27)

When the cubic axes are chosen as coordinate axes one find

wan(m)= (κ0 + κ1) +κ1(m12m2

2 +m22m3

2 +m32m1

2) . (6.28)

When κ1>0 there are six equivalent energy minima corresponding to the directionsm∈±u1,±u2,±u3(see the right of Figure 6.3). Conversely, when κ1 < 0 a more complex situation arises. In fact, there

are eight equivalent minima along the directions pointing the vertices of the cube (e.g. the direction

[1, 1, 1]) and the coordinate axes directions become now hard axes. The constants κ0 only ensure that

the anisotropy energy remains positive in both cases. The constants κ0 and κ1 are both functions of T .

Finally, for hexagonal crystals

wan(m)=κ1|m×u|2 + κ2|m×u|4, (6.29)

where the hexagonal axis is directed along the unit vector u. The constants κ0 only ensure that the

anisotropy energy remains positive in both cases. The constants κ0 and κ1 are both functions of T .

It is important to underline that the character of anisotropy interaction is pointwise in nature. In other

terms the value of the anisotropy energy density wan(m(x)) at a point x of the ferromagnetic specimen

occupying the region Ω, depends only on the direction of the magnetization m(x) at x∈Ω.

6.3.3 The Exchange energy term

As already mentioned, in spite of its great success in explaining the temperature behavior of the magneti-

zation, Weiss theory is silent on the physical origin of the molecular field. In 1928, Heisenberg showed that

the strong tendency that magnetic moments have to align into a common direction is due to an entirely

Micromagnetics

78

quantum mechanical effect (without classical analogue) which he called the exchange interaction.

Roughly speaking, the exchange interaction between two electrons results from both the necessity of the

wave-function describing the state of the system to be antisymmetric with respect to particle interchange,

as well as from their mutual electrostatic interaction.

A detailed analysis shows that if two atoms i and j have unpaired electrons, the exchange Hamiltonian

describing the exchange energy of two nearest-neighbor spins is given by

Wij :=−2JijSi ·Sj , (6.30)

~Si is the spin angular momentum of spin i, and Jij is an integral, called the exchange integral, which

depends on the overlapping spatial portion of the two wave-functions describing the set of unpaired

electrons belonging to each atom. The exchange integral Jij is positive for ferromagnetic materials.

The exchange energy is a short-range interaction; it is important only for neighboring spins and

decreases very rapidly with the distance, and has a minimum when the spins are oriented in the same

direction. Assuming that the angle θij between Si and Sj is very small, the two spins may be considered

as classical angular momentum vectors and one may write

Si ·Sj=S2cos θij=∼S2[

1− 1

2θij2]

=S2

(

1− 1

2|vj− vi|2

)

(6.31)

where vi is the unit vector along −Si (and therefore along the associated magnetic moment), and where

S= |Si|= |Sj |. We assume further that vi may be approximated sufficiently with a continuous function v

of position; then v can be interpreted macroscopically as the unit vector along M, i.e., v=m. Moreover

we assume that J := Jij is constant with respect to i and j. If ryx is the position vector of spin y with

respect to spin x, then vy − vx =m(y) −m(x) = ryx · ∇m(x) in the approximation assumed, and the

excess energy due to the non-parallelism of Sx and Sy is

Wij=∼−2JS2

(

1− 1

2|m(y)−m(x)|2

)

=−2JS2 + JS2|ryx · ∇m(x)|2. (6.32)

If there are n spins per unit volume, neglecting the constant term, we finish with the following density

of the excess exchange energy

wex(x) :=1

2nJS2

∑

y=/ x

|ryx · ∇m(x)|2 , (6.33)

where the sum is over nearest neighbors.

aa a

Figure 6.4. The expression of the exchange energy density wex can be simplified once the geometry of the latticeis known. (left) Simple cubic crystal (c=1). (center)Face centered cubic crystal (c=2). (right)Hexagonal crystal.


79

The previous relation can be simplified if the geometry of the lattice is known. To this end we observe

that:

|ryx · ∇m(x)|2 = 2∑

i=1

3

(ryx · ei)(ryx · ei+1)∂im(x) · ∂i+1m(x)

+∑

i=1

3

(ryx · ei)2|∂im(x)|2,(6.34)

and therefore, changing the order of summation

wex(x) =nJS2

(

∑

y=/ x

(ryx · ei)(ryx · ei+1)

)

∑

i=1

3

∂im(x) · ∂i+1m(x)

+1

2nJS2

(

∑

y=/ x

(ryx · ei)2)

|∇m|2(6.35)

For a cubic crystal (see Figure 6.4), it is possible to show

∑

y=/ x

(ryx · ei)(ryx · ei+1) =0 ,∑

y=/ x

(ryx · ei)2 =1

3

∑

y=/ x

|ryx|2 (6.36)

and that the quantities∑

y=/ x|ryx|2 is independent of x. Therefore

wex(x)=1

2Aex(|∇m1|2 + |∇m2|2 + |∇m3|2) (6.37)

with

Aex :=1

3nJS2

∑

y=/ x

|ryx|2 =2JxyS2 c

a(6.38)

where a is the length of the edge of the unit cell, and where c=1, 2, and 4 for simple body-centered, and

face-centered cubic lattices, respectively.

For a hexagonal crystal (see Figure 6.4), it is possible to show that

wex(x)=1

2Aex,1(|∂1m|2 + |∂2m|2)+

1

2Aex,2|∂3m|2 (6.39)

where Aex,1 and Aex,2 are to constant quantities given by

Aex,1 =1

2nJS2

∑

y=/ x

ρyx2 , Aex,2 =nJS2

∑

y=/ x

zy2 ; (6.40)

ρyx is the projection of ryx in the basal plane, and zy is its projection along the hexagonal axis. For ideal

close packing, as in cobalt, this reduces again to (6.37), with

Aex= 4nJS2a2 = 4 2√

JS2/a; (6.41)

a being the distance between nearest neighbors. Thus formula (6.37) covers most of the cases of interest.

Micromagnetics

80

As before, we now interpret (6.37) as a formula for the contribution of the non-uniformity of magneti-

zation to the exchange term in the free-energy density at any temperature, with Aex possibly temperature-

dependent. The constant Aex is known as the exchange stiffness constant; its value in ferromagnets

is usually of the order of 10−11Jm−1.

6.4 The Gibbs-Landau free energy functional GL.

On collecting all terms in the Helmholtz free energy functional F and adding the term that transforms

F to the Gibbs-Landau free energy functional GL, we get for a ferromagnet occupying the region Ω

GL(T ,M)=

∫

Ω

1

2Aex|∇m|2 +wan(m)− 1

2µ0Hd[M] ·M− µ0H0 ·Mdτ (6.42)

Here ϕan, has the form (6.27) or (6.29), as may be appropriate. Moreover,m :=M/Ms(T ) where Ms(T )

is the spontaneous magnetization at the given temperature T . Indeed, as already pointed out, when

the temperature is well below the Curie temperature of the ferromagnet, the strong exchange interaction

prevails over all other forces at the smallest spatial scale compatible with the continuum hypothesis. This

fact is taken into account by imposing the fundamental micromagnetic constraint:

|M|=Ms(T ) (6.43)

which means that, although the direction of M is in general nonuniform, i.e., it varies from point to

point, the magnitude of the local magnetization vector at each point inside the ferromagnet is equal to

the spontaneous magnetization Ms(T ) [BMS09].

6.4.1 The Gibbs-Landau free energy functional GL in normalized form.

It is mathematical useful and physical insightful to rewrite the micromagnetic free energy, in normalized

form[BMS09], where magnetization M and fields are measured in units of Ms, while energies are measured

in units of µ0Ms2|Ω|, where |Ω| is the volume of the ferromagnet. Then, magnetization states are described

by the unit vectorm :=M/Ms, and the fundamental micromagnetic constraint (6.43) assumes the form:

|m|= 1 in Ω. The normalized free energy GL associated with the vector field m is then given by:

GL(m,h0)=1

|Ω|

∫

Ω

ℓex2

2|∇m|2 + ϕan(m)− 1

2hd[m] ·m−h0 ·m dτ (6.44)

where

ℓex2 :=

Aex

µ0Ms2 , ϕan(m) :=

wan(m)

µ0Ms2 , hd[m] :=

Hd[M]

Ms, h0 :=

H0

Ms. (6.45)

The temperature dependent constant ℓex is the so-called exchange length. The function ϕan is now

referred to as the normalized anisotropy energy density, while hd[m] and h0 are referred to as the

normalized demagnetizing field and the normalized applied field. This free energy GL depends

on the magnetizationm, the applied magnetic field h0, and the temperature T . We omit the dependence

of GL from T , since in the subsequent discussion the temperature will always be assumed to be uniform

in space and constant in time.

In all subsequent chapters, the variational problems arising from the study of the Gibbs-Landau free

energy functional, will be studied by using the normalized form of GL as expressed by (6.44).

6.4 The Gibbs-Landau free energy functional GL.

81

7Equilibria of GL. Brown’s Equations.

In this chapter we investigate the existence of minimizers of the Gibbs-Landau free energy functional

GL(m) =+ℓex2

2|Ω|‖∇m‖Ω2 − 1

2|Ω|(hd[m],m)Ω + ‖ϕan(m)‖Ω2 − 1

|Ω|(h0,m)Ω (7.1)

Let us make the statement more precise. The class of admissible maps we are interested in is defined as

H1(Ω, S2) := m∈H1(Ω,R3) : m(x)∈ S2 for τ -a.e. x∈Ω,

where we have denoted by τ the Lebesgue measure on R3, and by S

2 the unit sphere of R3. We consider

H1(Ω,S2) as a metric space endowed with the metric structure induced by the classical H1(Ω,R3) metric.

The anisotropy density energy ϕan2 : S

2 →R+ is supposed to be a globally lipschitz function, i.e.

∃κL> 0 such that

|ϕan2 (m1)− ϕan

2 (m2)|6 κL|m1−m2| ∀m1,m2∈S2. (7.2)

The hypothesis assumed on ϕan2 is sufficiently general to treat the most common cases of crystal anisotropy

energy densities arising in applications. As a sake of example, for uni-axial anisotropy, the energy density

reads

ϕan2 (m)= κ(y)[1− (u ·m)2], (7.3)

the unit vector u being the easy axis of the crystal. For cubic type anisotropy, the energy density reads as:

ϕan2 (m) =κ

∑

i=1

3

[(ui ·m)2− (ui ·m)4] (7.4)

the unit vectors ui being the easy three mutually orthogonal axes of the cubic crystal.

7.1 The existence of minimizers for GL.

In this section we prove the following

Proposition 7.1. The set H1(Ω, S2) is weakly sequentially closed in H1(Ω,R3). Moreover the Gibbs-

Landau free energy functional GL is coercive and weakly lower semicontinuous on H1(Ω,R3); there thus

exists at least a minimizer of GL in H1(Ω, S2).

83

Proof. To say that H1(Ω, S2) is weakly closed means that

[(mn)n∈N∈H1(Ω, S2)m∞] =⇒ [m∞∈H1(Ω, S2) ]. (7.5)

To prove the previous implication we recall that the injection ι: H1(Ω, R3) → L2(Ω, R3) is compact

(Rellich-Kondrachov theorem) and therefore the injection operator ι maps weakly convergent sequences

of H1(Ω,R3) into strongly convergent sequences of L2(Ω,R3). Thus we have

limn→∞

mn=m∞ in L2(Ω,R3).

Moreover there exists a subsequence (mnk)k∈N such that (mnk

)k∈N →m∞ a.e. in Ω. In particular for

a.e. x∈Ω one has

1= limk→∞

|mnk|= |m∞|,

and hence m∞∈H1(Ω, S2).

We now prove that GL is coercive. Indeed for some cG ∈R+ one has

infm∈H1(Ω,S2)

GL(m) 6 infu∈U(Ω,S2)

GL(u) 6ℓex2

2cG2 . (7.6)

Moreover if m0 is a global minimizer of GL than necessarily

ℓex2

2|Ω|‖m0‖H1(Ω,R3)2 6

ℓex2

2|Ω|‖m0‖Ω2 + GL(m0) 6

ℓex2

2( 1+ cG

2 )

and hence

infm∈H1(Ω,S2)

GL(m) = infm∈KG(Ω,S2)

GL(m) (7.7)

with

KG(Ω, S2) :=

m∈H1(Ω, S2) : ‖m‖H1(Ω,R3)2 6 |Ω|( 1 + cG

2 )

. (7.8)

We now observe that KG(Ω, S2) ⊆H1(Ω, S2) is weakly compact, being the intersection of a sequentially

weakly closed set H1(Ω, S2) with a weakly compact set (ball) of H1(Ω, R3). Eventually, we observe

that the unconstrained functional GL – defined in H1(Ω,R3) – is Frechet differentiable and convex, and

therefore weakly lower semicontinuous on H1(Ω,R3).

Remark 7.1. We want to explicitly observe that the same result still holds if no anisotropy energy

density and no Zeeman interaction energy is considered in the expression of GL.

7.2 A first glance to the local equilibria of GL. First order (external) variation of GL.

We start this section by recalling that, according to Micromagnetics, the local state of magnetization of

matter is described by a vector field, the magnetization m, defined over Ω which is the region occupied

by the body. The equilibrium states result in extrema of the Gibbs-Landau free energy functional GL,

which, in normalized units, is expressed by (6.44):

GL(m,h0) =1

|Ω|

∫

Ω

ℓex2

2|∇m|2 + ϕan(m)− 1

2hd[m] ·m−h0 ·m dτ (7.9)

Equilibria of GL. Brown’s Equations.

84

where m: Ω → S2 is a vector field taking values on the unit sphere S

2 of R3, |Ω| denotes the volume of

the region Ω, and ℓex2 is a positive material dependent constant. The constraint on the image of m being

due to the normalized version of the fundamental micromagnetic constraint (6.43), which assumes that

a ferromagnetic body well below the Curie temperature is always locally saturated. This means that the

following constraint is satisfied: |m|= 1 almost everywhere in Ω.

7.2.1 Weak Euler-Lagrange equation for GL: weak Brown’s static equation.

A natural space in which to look for minimizers of the Gibbs-Landau functional is one in which the energy

(7.9) is finite. Since the induced magnetic field operator hd has a meaning in L2(Ω,R3) (see Chapter 4),

and the exchange energy has a meaning in the Sobolev space H1(Ω,R3) we will assume m∈H1(Ω,R3)

and we will write m∈H1(Ω, S2) to emphasize that the magnetization field satisfies the local saturation

constraint given by |m|= 1 a.e. in Ω.

We recall that H1(Ω,R3) is the space of square summable vector fieldsm∈L2(Ω,R3) whose first order

weak partial derivatives ∂im belong to L2(Ω,R3). In terms of the L2(Ω,R3) norm and scalar product,

the Gibbs-Landau free energy functional can be written as

GL(m,h0) =ℓex2

2|Ω|‖∇m‖Ω2 − 1

2|Ω|(hd[m],m)Ω + 〈ϕan(m)〉Ω−h0 · 〈m〉Ω. (7.10)

The study of the equilibrium states conduces to the computation of the (external) first order variation of

GL, for an arbitrary variation u∈H1(Ω, S2) compatible with the fundamental micromagnetic constraint

(|m|=1), i.e. such that |m+u|2 =1 a.e. in Ω. To this end, for every ε>0, let us consider the subset of

H1(Ω,R3) given by

H1(Ω, Bε) := u∈H1(Ω,R3) : |u|<ε a.e. in Ω. (7.11)

Since |m+u|> |m| − |u|, we have that |m+ u|> 0 almost everywhere in Ω whenever m ∈H1(Ω, S2),

u ∈ H1(Ω, Bε) and |ε| < 1. Therefore for every |ε| < 1 it makes sense to consider the nearest point

projection operator defined by

πm:u∈H1(Ω, Bε) 7→ m+u

|m+u| ∈H1(Ω, S2) (7.12)

The operator πm is Frechet differentiable in H1(Ω, Bε), and a simple computation shows that

τ (u) := dπm(0)u=m× (u×m)=u− (u ·m)m (7.13)

for every u∈H1(Ω,R3). Now, if m∈H1(Ω, S2) is a stable equilibrium state for GL then

GL(m) 6 GL(πm(u)) (7.14)

for sufficiently small ε(m) and every u ∈ H1(Ω, Bε). Therefore, from the chain rule for the Frechet

differential [Ma02], we find that necessarily 〈dGL(πm(0)), dπm(0)u〉= 0 for every u∈H1(Ω,R3). Since

πm(0)=m, denoting the differential of πm at 0 by τ , we finish with the condition

〈dGL(m), τ (u)〉= 0 ∀u∈H1(Ω,R3). (7.15)

7.2 A first glance to the local equilibria of GL. First order (external) variation of GL.

85

Thus, denoting by Tm the tangent space to H1(Ω, S2) in m, i.e. the image space of τ , we finish with the

following weak Euler-Lagrange equation for the Gibbs-Landau free energy functional (also called

weak Brown’s static equation):

〈dGL(m),v〉= 0 ∀v ∈Tm. (7.16)

In this way we have shown that the role played by the fundamental micromagnetic constraint is to restrict

the differential of GL to the tangent space Tm. The only thing that remains to compute is the differential

of the unconstrained GL. A straightforward computations shows that

|Ω|〈dGL(m),u〉= ℓex2 (∇m,∇u)Ω− (hd[m],u)Ω + (∇ϕan(m),u)Ω− (h0,u)Ω (7.17)

for every u∈H1(Ω,R3). Therefore, wheneverm is an equilibrium state for GL, the restriction of dGL(m)

to Tm must necessarily vanish.

7.2.2 The regular case: Brown’s static equations.

When m,u∈C02(Ω,R3), relation (7.16) can be written as

−|Ω|〈dGL(m),u〉= (heff[m], τ (u))Ω = 0 (7.18)

for every u∈H1(Ω,R3), where we have denoted by

heff[m] := ℓex2 ∆m+hd[m] +han(m)+h0 (7.19)

the so-called normalized effective field, and by han(m) :=−∇ϕan(m) the normalized anisotropy

field. Since by cyclic permutation

(heff[m], τ (u))Ω = (heff[m],m× (u×u))Ω = (m× (m×heff[m]),u)= 0 (7.20)

for every u ∈ C02(Ω, R3), by the du Bois-Reymond lemma, one finish with the Brown’s static

condition for local equilibria

m× (m×heff[m]) =0 in Ω. (7.21)

Moreover, by taking m,u∈C2(Ω,R3), and taking into account (7.18) we finish with the condition

∫

∂Ω

∂nm · τ (u) dσ=0 (7.22)

for every u∈C2(Ω,R3), and therefore, following the same argument that leads to (7.21), we finish with

the boundary condition m × ∂nm= 0 on ∂Ω. However, since |m| = 1 on ∂Ω, one has m · ∂nm= 0 on

∂Ω, and hence 0 = |m × ∂nm|2 = |∂nm|2 − |m · ∂nm|2 = |∂nm|2. Therefore the boundary condition

m× ∂nm= 0 is equivalent to the Neumann boundary condition

∂nm= 0 on ∂Ω. (7.23)

Summarizing:


86

Proposition 7.2. Let Ω be an open connected subset of R3. If m∈H1(Ω, S2) is an equilibrium state

of GL then the weak Euler-Lagrange equation holds:

〈dGL(m),v〉= 0 ∀v ∈Tm, (7.24)

where we have denoted by Tm the tangent space to H1(Ω, S2) in m. Moreover, if m∈C2(Ω,R3) then

the Brown’s static equations hold:

m× (m×heff[m]) =0 in Ω∂nm=0 on ∂Ω .

(7.25)

where the effective field is given by heff[m] := ℓex2 ∆m+hd[m] +han(m) +h0.

The Brown’s static equations express the fact that the local torque exerted on the magnetization by

the effective field must be zero at equilibrium. The Neumann boundary condition ∂nm=0 is valid when

no surface anisotropy is present.

7.2.3 Brown’s static equations for uniform magnetizations.

In absence of an external applied field (h0=0), and when no volume and no surface anisotropy is present

(han(m) = 0), from (7.25) we get that, at equilibrium, every constant in Ω magnetization m must be

such that

m×hd[m] = 0 in Ω. (7.26)

In other terms, at equilibrium,m and hd[m] must be aligned in Ω. This can be mathematically expressed

by saying that m must be a generalized eigenvector of the demagnetizing field hd, i.e. the constant

in Ω magnetization m must be such that for some function λ∈C∞(Ω,R3)

hd[m] =λm in Ω. (7.27)

Since for ellipsoidal particles, every constant in Ω magnetization, directed along one of the principal axes

of the ellipsoid, is a classical eigenfunction of hd in Ω (see Chapter 5), we reach the conclusion that, at

equilibrium, m must be directed along one of the principal axes of the ellipsoid too. Which one of the

axes, is a question that depends on the type of the equilibrium states (minimum, maximum or saddle),

and can be investigated by the means of the second order (external) variation.

7.3 A first glance to the local minimizers of GL. Second order (external) variation of GL.

It is important to stress that Brown’s equation (7.25) determines all possible magnetization equilibria

regardless of their stability. However, according to the thermodynamic principle of free energy mini-

mization, only GL minima will correspond to stable equilibria and, thus, will be in principle physically

observable. The information on the nature of equilibria can be obtained by computing the second variation

of GL and determining if it is positive under arbitrary variations of the vector field m, subject to the

fundamental constraint |m|= 1 a.e. in Ω.


87

Aim of this section is the computation of the second order (external variation) of GL which, as we

will see soon, plays an important rule in the identification of the direction of magnetization of the local

minimizers of the Gibbs-Landau free energy functional. The computations, is straightforward if based on

the higher order chain rule for the Frechet differential [Ma02], and conduce to the expression

|Ω|〈d2GL(m),v2〉 =ℓex2

2[‖∇v‖Ω

2 −‖|v |∇m‖Ω2 ]

1

2[(hd[v],v)Ω

2 + (hd[m], |v |2m)Ω]

+[(D2ϕan(m)v ,v)Ω−∥

∥

∥|v |ϕan1/2(m)

∥

∥

∥

Ω

2 ]−(h0, |v |2m)Ω ,

(7.28)

for every v ∈Tm. Thus:

Proposition 7.3. Let Ω be an open connected subset of R3. If m∈H1(Ω, S2) is a local minimizer

of GL then

〈d2GL(m),v2〉> 0 ∀v ∈Tm, (7.29)

where we have denoted by Tm the tangent space to H1(Ω, S2) in m.

In absence of an external applied field (h0 = 0), and when no volume and no surface anisotropy is

present (han(m) = 0), from (7.28) and (7.29), we get that constant in Ω local minimizer m must be

such that

−(hd[m],m)Ω 6−(hd[v],v)Ω|v |2 (7.30)

for every constant in Ω tangential variation v ∈Tm.

Figure 7.1. Tri-axial ellipsoid with distinct semi-axes lengths c > b > a.


88

For ellipsoidal particles (see Chapter 5) of semi-axes lengths c> b>a (see Figure 7.1), the condition

(7.30) amounts to require that the local minimizerm must be directed along one of the two major semi-

axes of the ellipsoid, i.e. m=±c/|c|. From similar considerations we then find that whenm is a constant

in Ω local maximizer of GL, m must be directed along one of the two minor semi-axes of the ellipsoid,

i.e. m := ±a/|a|. Finally, it can be shown that the states m := ±b/|b| are metastable magnetization

states of GL.


89

8Global Minimizers of GL.

In this chapter we introduce the Brown’s fundamental theorem on fine ferromagnetic particles, and

extend it to the case of a general ellipsoid [Fra11]. By means of Poincaré inequality for the Sobolev space

H1(Ω,R3), and some properties of the demagnetizing field operator, it is rigorously proven that for an

ellipsoidal particle, with diameter d, there exists a critical size (diameter) dc such that for d < dc the

uniform magnetization states are the only global minimizers of the Gibbs-Landau free energy functional

GL. A lower bound for dc is then given in terms of the demagnetizing factors.

8.1 Introduction

Theoretical discussions of the coercivity of magnetic materials make considerable use of the following

idea [Bro68]: «whereas a ferromagnetic material in bulk (in zero applied field) possesses a domain struc-

ture, the same material in the form of a sufficiently fine particle is uniformly magnetized to (very near) the

saturation value, or in other words consists of a single domain». But as Brown points out in [Bro68]: «the

idea as thus expresses, scarcely is to be called a theorem, for it is not a proved proposition nor a strictly

true one».

The first rigorous formulation of this idea is due to Brown himself who, in his fundamental

paper [Bro68] rigorously proved for spherical particles what is known as Brown’s fundamental

theorem of the theory of fine ferromagnetic particles. This fundamental theorem states the

existence of a critical radius rc of the spherical particle such that for r < rc and zero applied field

the state of lowest free energy (the ground state) is one of uniform magnetization.

The physical importance of Brown’s fundamental theorem is that it formally explains, although in

the case of spherical particles, the high coercivity that fine particles materials have, compared with

the same material in bulk [Bro68]. In fact, if the particles are fine enough to be single domain, and

magnetic interactions between particles have a negligible effect, each individual particle can reverse its

magnetization only by rigid rotation of the magnetization vector of the particle as a whole, a process

requiring a large reversed field (rather than by domain wall displacement, which is the predominant

process in bulk materials at small fields) [Bro68].

The main limitation of the theorem is that it is applicable to spherical particles whereas, real particles

are most of the time elongated [Aha88]. Motivated by this, Aharoni [Aha88], by using the same math-

ematical reasoning as Brown, was able to extend the Fundamental Theorem to the case of a prolate

spheroid. The main objective of this paper is to extend, by means of Poincaré inequality for the Sobolev

space H1(Ω,R3) [PW60, Beb03] and some properties of the magnetostatic self-energy [Bro62b, Bro63,

91

Fri80, Aha91], the fundamental theorem of Brown to the case of a general ellipsoid. In the sequel,

it is rigorously proven that for an ellipsoidal particle, with diameter d, there exists a critical size

(diameter) dc such that for d<dc the uniform magnetization states are the only global minimizers of the

micromagnetic free energy functional. A lower bound for dc is then given in terms of the demagnetizing

tensor eigenvalues [Sim95] (the so called demagnetizing factors [Osb45]), which completely characterize

the induced magnetic field inside ellipsoidal particles, thanks to Payne and Weinberger result on the best

Poincaré constant [PW60, Beb03].

8.2 Formal theory of micromagnetic equilibria

In what follows, for the sake of clarity, we will neglect any anisotropy energy term in the expression of

the Gibbs-Landau functional (8.1). By the way, it is straightforward to extend our considerations to the

case when (for example) uni-axial anisotropy of the easy-axis type is present.

We start our discussion by recalling basic facts about micromagnetic theory. According to Micro-

magnetics the local state of magnetization of matter is described by a vector field, the magnetizationm,

defined over Ω which is the region occupied by the body. The stable equilibrium states of magnetization

are the minimizers of the so called Gibbs-Landau free energy functional associated with the magnetic

body. In dimensionless form, in zero applied field, and when no anisotropy energy density is present, this

functional can be written as (see §6.4.1):

GL(m,Ω) =1

|Ω|

∫

Ω

(

ℓex2

2|∇m|2− 1

2hd[m] ·m

)

dτ , (8.1)

where m: Ω→ S2 is a vector field taking values on the unit sphere S

2 of R3, and |Ω| denotes the volume

of the region Ω, and ℓex2 is a positive material constant.

The constraint on the image of m is due to the following fundamental assumption of the micromag-

netic theory: a ferromagnetic body well below the Curie temperature is always locally saturated. This

means that the following constraint is satisfied:

|m|= 1 a.e. in Ω. (8.2)

Global micromagnetic minimizers correspond to vector fields which minimize the Gibbs-Landau energy

functional (8.1) in the class of vector fields which take values on the unit sphere S2.

8.3 The magnetostatic self-energy. Mathematical properties of the dipolar magnetic field. TheBrown lower bound

The energy functional GL given by (8.1) is the sum of two terms: the exchange energy and the Maxwellian

magnetostatic self-energy (the second term). The magnetostatic self-energy is the energy due to the

(dipolar) magnetic field hd[m] generated by m. From the mathematical point of view, assuming Ω to

be open, bounded and with Lipschitz boundary, and denoting with χΩm the trivial extension of the

magnetization m to all the space R3, the induced magnetic field can be defined as the unique vector

field hd[m] ∈ L2(R3, R3) which satisfies (in the sense of distributions on R3) the following Maxwell’s

equations [Fri80, Sim93]:

div (hd[m] +mχΩ) =0curl hd[m] =0 .

(8.3)

Global Minimizers of GL.

92

We recall that the operator hd which to every m∈L2(Ω,R3) associate the unique solution hd[m] of the

above Maxwell’s equations, is a bounded, self-adjoint and negatively semi-defined linear operator with

‖hd‖op= 1, when endowed with the L2(Ω,R3) scalar product given by

(m,u)Ω =

∫

Ω

m ·udτ . (8.4)

Self-adjointness means that (hd[m], u)Ω = (m, hd[u])Ω for every m, u ∈ L2(Ω,R3), while semi-definite

negativeness states that, for every m∈L2(Ω,R3), we have

−(hd[m],m)Ω≥ 0. (8.5)

Obviously the semi-definite positiveness of the induced magnetic field assures the positiveness of the

Gibbs-Landau free energy functional.

Finally let us recall the following Brown lower bound to the magnetostatic self-energy [Bro62a] as

reported by himself in [Bro68]. Consider an arbitrary irrotational vector field h which is defined over the

whole space R3 and is regular at infinity. Under these assumptions Brown proved that:

−∫

Ω

h ·m dτ − 1

2

∫

R3

|h|2 dτ ≤−1

2

∫

Ω

hd[m] ·mdτ , (8.6)

the equality holding if and only if h=hd[m]. In other terms, for every irrotational and regular at infinity

vector field h: R3 → R3, the left hand side of (8.6) does not exceed the magnetostatic self-energy and

becomes equal to it only when h is everywhere equal to hd[m]. It is worthwhile emphasizing that the

vector field h in this inequality needs not be related in any way to m [Aha91]. A very useful particular

case of this lower bound can be obtained by letting h=hd[u] with u∈L2(Ω,R3). In this way we arrive

at the following form of the Brown lower bound which we state here as a lemma:

Lemma 8.1. Let Ω⊆R3 be open, bounded and with Lipschitz boundary. For every u,m∈L2(Ω,R3):

−(hd[u],m)Ω +1

2(hd[u],u)Ω≤−1

2(hd[m],m)Ω, (8.7)

with equality if and only if u=m.

8.4 The case of ellipsoidal geometry. Demagnetizing tensor

Since hd is a linear operator, the restriction of hd to the subspace U(Ω,R3) of constant in space vector

fields can be identified with a second order tensor known as the effective demagnetizing tensor of Ω

and defined by [Sim95, Osb45]:

Neff[m] =−∫

Ω

hd[m] dτ =−|Ω|〈hd[m]〉Ω , (8.8)

8.4 The case of ellipsoidal geometry. Demagnetizing tensor

93

where m∈U(Ω,R3) and for all u∈L2(Ω,R3) we have denoted with

〈u〉Ω =1

|Ω|

∫

Ω

udτ (8.9)

the average of u over Ω. The tensor Neff is known in literature as the effective demagnetizing tensor

of Ω, where the qualifier effective is used as a reminder of the fact that Neff is related to the average of

hd[m] over Ω [Sim95, Osb45]. In addition to that, as we have already seen in Chapter 5, when Ω is an

ellipsoid andm∈U(Ω,R3) also hd[m]∈U(Ω,R3); i.e. if Ω is an ellipsoid andm is constant, then hd[m]

is also constant in Ω. In physical terms this means that uniformly magnetized ellipsoids induce uniform

magnetic fields in their interiors. In this case the effective demagnetizing tensor Neff is pointwise related

to m since the relation (8.8) becomes:

Neff[m] =−hd[m]. (8.10)

In the rest of the present paper we will indicate with Nd the demagnetizing tensor associated to an

ellipsoidal particle Ω. Obviously from (8.5) we get that the quadratic formQd(m)=Nd[m] ·m is a positive

semi-definite quadratic form. We will indicate with

µ2 := infu∈R3\0

Qd(u)

|u|2 (8.11)

the first eigenvalue associated to this quadratic form, i.e. the minimum demagnetizing factor for the

general ellipsoid Ω. This quantity can be expressed analytically in terms of elliptic integrals (see Chapter

5).

Remark 8.1. It is important to stress that the eigenvalues of the quadratic form Qd are shape-dependent

but not size-dependent so that, when the volume |Ω| is changed by preserving the shape of the ellipsoid,

µ2 does not change.

8.5 The exchange energy and the Poincaré inequality. Null average micromagnetic minimizers

The exchange energy (the first term in eq. (8.1)), energetically penalize spatially non-uniform magne-

tization states: it takes into account the presence of the microscopic exchange interactions which tends

to align the atomic magnetic moments.

A natural space in which to look for minimizers of the Gibbs-Landau functional is one in which the

energy (8.1) is finite. Since the induced magnetic field operator hd has a meaning in L2(Ω,R3), and the

exchange energy has a meaning in the Sobolev space H1(Ω,R3) we will assume m ∈H1(Ω,R3) and we

will writem∈H1(Ω,S2) to emphasize that the magnetization field satisfies the local saturation constraint

given by |m|= 1 a.e. in Ω.

We recall that H1(Ω,R3) is the space of square summable vector fieldsm∈L2(Ω,R3) whose first order

weak partial derivatives ∂im belong to L2(Ω,R3). We also recall that in the Sobolev space H1(Ω,R3)

the following Poincaré inequality holds [PW60, Beb03]:


94

Lemma 8.2.Let Ω be a bounded connected open subset of R3 with a Lipschitz boundary. Then there

exists a constant cP (the so called Poincaré constant), depending only on Ω, such that for every vector

field m∈H1(Ω,R3):

‖m−〈m〉Ω‖Ω≤ cp‖∇m‖Ω (8.12)

where 〈m〉Ω denotes the spatial average of m over Ω (see eq. (8.9)).

For practical purposes is important to know an explicit expression for the Poincaré constant. The main

result is this direction concerns the special case of a convex domain [PW60, Beb03].

Lemma 8.3. Let Ω be a convex domain with diameter diam(Ω). Then for every vector fieldm∈H1(Ω,

R3):

‖m−〈m〉Ω‖Ω≤ diam(Ω)

π‖∇m‖Ω. (8.13)

In terms of the L2(Ω,R3) norm and scalar product the Gibbs-Landau functional (8.1) reads as:

GL(m,Ω)=ℓex2

2|Ω|‖∇m‖Ω2 − 1

2|Ω|(hd[m],m)Ω. (8.14)

We now observe that if m0∈H1(Ω,S2) is a global minimizer of the Gibbs-Landau energy functional (8.14)

then for every u∈U(Ω,R3) such that |u|=1 a.e. in Ω, we have ‖∇u‖Ω2 = 0. Thus

GL(m0) ≤ − 1

2|Ω|(hd[u],u)Ω, (8.15)

and hence:

GL(m0) ≤ 1

2inf

|u|=1Qd(u) =

1

2µ2. (8.16)

From this simple observation and the use of Poincaré inequality (8.12) we get that if m0 is a null average

magnetization state, then

µ2≥ 2 GL(m0)≥ ℓex2

cP2 (8.17)

and hence cP ≥ ℓex µ−1. Thus we proved the following lemma:

Lemma 8.4. Let Ω ⊆ R be an ellipsoid and let m0 ∈ H1(Ω, S2) be a global minimizer of the Gibbs-

Landau energy functional (8.1). If 〈m0〉Ω =0 then

diam(Ω)≥ π ℓexµ

(8.18)

where we have indicated with diam(Ω) the diameter of the ellipsoid Ω.

8.5 The exchange energy and the Poincaré inequality. Null average micromagnetic minimizers

95

We recall that diam(Ω) is defined as the largest distance between couples of points in Ω, and in the case

of an ellipsoid it coincides with two times the largest semi-axis. By letting |Ω| decrease by keeping the

shape of ellipsoid invariant, so that µ is constant, we arrive to a violation of the the inequality (8.18)

which implies that zero-average global minimizers cannot exist when the dimension of the particle is

reduced below the critical diameter π ℓex µ−1. From the physical point of view, this result is interesting

in its own right when one interprets zero-average global minimizers as the usual demagnetized states of a

magnetic particle. The above Lemma implies that there is no unmagnetized ground state in fine particles.

8.6 The generalization of the fundamental theorem of Brown to the case of ellipsoidal particles

Consider an homogeneous ferromagnetic particle occupying the region of space Ω which is assumed to

be a general ellipsoid in R3 and let m∈H1(Ω, S2). From (8.7) we have that for every constant in space

vector field u∈U(Ω,R3):

|Ω|Nd[u] · 〈m〉Ω− 1

2|Ω|Qd(u)≤−1

2(hd[m],m)Ω. (8.19)

In particular, letting u= 〈m〉Ω we get that for all m∈L2(Ω,R3):

|Ω|Qd(〈m〉Ω)≤−(hd[m],m)Ω. (8.20)

From Lemma 8.5.4 we get that if cP < ℓex µ−1 then the global minimizer m0 cannot be null average

(〈m0〉Ω =/ 0) and so after dividing and multiplying the left hand side of (8.20) by |〈m0〉Ω|2, passing to the

inf we get:

|〈m0〉Ω|2 µ2≤−(hd[m],m)Ω. (8.21)

From (8.16) and (8.21) we infer that if m0 is a global minimizer for GL then:

µ2≥ 2 GL(m0)≥ ℓex2

cP2 (1− |〈m0〉Ω|2)+ µ2|〈m0〉Ω|2, (8.22)

where the first lower bound is due to Poincaré inequality (8.12). Thus we arrive to the conclusion that

if m0 is a global minimizer for GL then:

(1− |〈m0〉|2)(

ℓex2

cP2 − µ2

)

≤ 0. (8.23)

As a consequence if cP < ℓex µ−1, then |〈m0〉|2 = 1 and hence m0 is constant a.e. in Ω. We have in this

way proved the following generalization of Brown’s fundamental theorem for fine ferromagnetic particles:

Theorem 8.5. Let Ω⊆R3 be an ellipsoid and let m0∈H1(Ω, S2) be a global minimizer of the Gibbs-

Landau free energy functional (8.1). If cP <ℓex µ−1 then m0∈U(Ω,R3), i.e. m0 is constant a.e. in Ω.

Thus a sufficient condition for m0 to be constant is that

diam(Ω)<π ℓex µ−1 (8.24)

where diam(Ω) is twice the largest semi-axis of the ellipsoid Ω.

The inequality (8.24) means that if we consider particles of given ellipsoidal shape (given ratio of semi-

axes) with decreasing volume, there is a critical dimension below which the global minimizers (ground

states) are uniform.


96

β> γ

β= 0.05

β= 0.075

β= 0.1β= 0.15

β= 0.2

ac

ℓex

35

30

25

20

15

10

5

0

γ 0.05 0.1 0.15 0.2

Figure 8.1. Plot of the lower bound ac as a function of the ellipsoid semi-axes aspect ratios β and γ.

It is interesting to consider the case of very slender ellipsoid, i.e. an ellipsoid with semi-axes a≫ b≥ c.In this case, the asymptotic behavior of µ2 is given by [Osb45]:

µ2≈ b c

a2

[

log

(

4 a

b+ c

)

− 1

]

. (8.25)

Now, by using the notation β = b/a and γ = c/a, and the fact that diam(Ω) = 2 a, the inequality (8.24)

can be read as

a<ac=π

2ℓex

1

β γ√

[

log

(

4

β+ γ

)

− 1

]

−1/2

, (8.26)

which provides a more explicit lower bound for the critical size to have spatially uniform ground in

ellipsoidal particles. In Fig. 8.1 we report the behavior of the lower bound ac as a function of the ellipsoid

semi-axes aspect ratios β and γ, computed from eq. (8.26).

8.7 Some remarks on the value of the critical size. The best Poincaré constant in the case ofa spherical particle

It is well known that the best Poincaré constant in H1(Ω,R3), in the class of all convex domains having

the same diameter, is given by cP =diam(Ω)/π. However it is also well known that once fixed the domain

Ω (not just the diameter), the best Poincaré constant is given by cP =λ1−1 where λ1 is the smallest positive

eigenvalue associated with the following Neumann problem for the Helmholtz equation:

−∆ ϕ=λϕ in Ω∂nϕ= 0 on ∂ Ω

. (8.27)

8.7 Some remarks on the value of the critical size. The best Poincaré constant in the case of a spherical particle

97

Thus a better estimate of (8.24) can be obtained by solving equations (8.27) when the geometry of Ω is

that of the general ellipsoid under consideration.

For the case of a spherical particle (a ball of radius r) the first eigenvalue of (8.27) is given by λ1=x11

r,

where x11 is the first positive root of the equation:

2 xJ1+

1

2

′ (x)−J1+1

2

(x)= 0,

and where we have indicated with Jα the Bessel functions of the first kind [Pol02, Liz01]. Equivalently

the factor x11 can be found computing the first positive root of the equation: j1′(x) = 0, where we have

indicated with j1 the spherical Bessel function, related to Jα by the equation:

j1(x) = (2x/π)−1/2 J1+1

2

(x). (8.28)

A numerical computation gives for this first positive root the approximated value x11 ≈ 2.0816. Thus

recalling that in the case of a sphere (see Chapter 5):

µ2 = inf|u|=1

Qd(u)=1

3, (8.29)

we get, from Theorem 8.6.5, that m0 is constant in space when cP =r

x11< ℓex µ

−1, and this inequality

holds if and only if:

r < rc , rc≈ 3.6055 ℓex. (8.30)

Thus, for the special case of of a spherical particle, we arrive to the same estimate found by Brown[Bro68].

8.8 Final considerations

We have extended the Brown’s fundamental theorem on fine ferromagnetic particles to the case of a

general ellipsoid, and given (by means of Poincaré inequality for the Sobolev space H1(Ω, R3)) an

upper bound to the critical size (diameter) under which the uniform magnetization states are the only

global minimizers of the Gibbs-Landau free energy functional GL. Although for the sake of clarity we

have neglected any anisotropy energy term in the expression of the Gibbs-Landau functional (8.1), it is

straightforward to extend the result to the case when (for example) uni-axial anisotropy of the easy-axis

type is present. The extension of this result to the case of local minimizers of the Gibbs-Landau functional

will be the aim of Chapter 9.


98

9Local Minimizers of GL

This chapter is devoted to the presentation of the results obtained in collaboration with Prof. François

Alouges and Prof.Benoit Merlet during my first PhD internship at CMAP, Ecole Polytechnique, Palaiseau

(Paris).

Precisely, the chapter concerns the study of local minimizers of the micromagnetic energy in small

ferromagnetic 3d convex particles for which we justify the Stoner-Wohlfarth approximation: given a

uniformly convex shape Ω⊂R3, there exist δc> 0 such that for 0< δ ≤ δc any local minimizer m of the

micromagnetic energy in the particle δΩ satisfies ‖∇m‖L2(δΩ)6Cδ2. In the case of ellipsoidal particles we

strengthen this result by proving that, for δ small enough, local minimizers are exactly spatially uniform.

This last result extends W.F. Brown’s fundamental theorem for fine 3d ferromagnetic particles [Brown

(1968), Di Fratta et al. (2011)] which states the same result but only for global minimizers.

As a by-product of the method that we use, it is also established a new Liouville type result for locally

minimizing p-harmonic maps with values into a closed subset of a Hilbert space. Namely, we establish

that in a smooth uniformly convex domain of Rd any local minimizer of the p-Dirichlet energy (p > 1,

p=/ d) is constant.

9.1 Introduction and main results

Micromagnetism as introduced by Landau-Lifshitz and Brown describes the magnetization states inside a

ferromagnetic body below the Curie temperature (see [LL35, Bro62b, Bro63]). According to this theory,

the magnetization in a ferromagnetic sample occupying the domain Ω ⊂ R3 is modeled by a vector

field m: Ω → R3, of constant magnitude Ms, the saturation magnetization, that we assume equal to 1

after normalization. The (static) theory then states that observed magnetization distributions are local

minimizers of the micromagnetic energy

E(m; Ω) : =ℓex2

2

∫

Ω

|∇m|2 dτ +

∫

Ω

ψ(m) dτ − 1

2

∫

Ω

hd[m,Ω] ·m dτ. (9.1)

99

The first term is called the exchange energy and ℓex is the exchange length. This term penalizes brutal

variations of the magnetization. The second term combines anisotropy effects and the action of an external

field ha

ψ(u)=Aanis(u)−ha ·u, (9.2)

where Aanis: S2 → R

+ is a non negative function that vanishes at the so-called easy directions . When

merged with the energy due to the external field, the corresponding contribution favors directions of mag-

netization which minimize ψ. The last term, called stray field energy is a non local self-interaction energy.

The vector field hd[m,Ω] (usually called stray field or demagnetizing field) represents the magnetic field

generated by magnetization distribution m itself through Maxwell equations. From a mathematical

point of view the simplest and shortest way to define hd[m,Ω] is to extend m by 0 in R3 \Ω by setting

m0 :=1Ωm. The stray field is then defined as the opposite of the projection of m0 in L2(R3,R3) on the

closed subspace,

V := ∇v : v ∈D ′(R3,R), ∇v∈L2(R3,R3). (9.3)

Existence of a minimizer of the micromagnetic energy is easily obtained by the direct method of the

calculus of variation (at least when ψ is lower semi-continuous and Ω⊂R3 is a non empty open set with

finite volume).

Here, we are interested in the behavior of the magnetization when the shape of the ferromagnetic

sample is fixed and its size is comparable to the exchange length. For this, we introduce a reference

domain Ω with unit diameter that we rescale by setting Ωδ := δΩ where δ > 0 is a (small) parameter.

Similarly, for any magnetization distributionmδ defined in the physical domain Ωδ we setm(x)=mδ(δx)

for x ∈ Ω, the reference domain. With this change of variable, we have δ∇mδ(δx) = ∇m(x) while

hd[mδ; Ωδ](δx)=hd[m; Ω](x). Therefore the three energy terms scale as

∫

Ωδ

|∇mδ|2 dτ = δ

∫

Ω

|∇m|2 dτ ,∫

Ωδ

ψ(mδ) dτ = δ3∫

Ω

ψ(m) dτ∫

Ωδ

hd[mδ; Ωδ] ·mδ dτ = δ3∫

Ω

hd[m; Ω] ·mdτ .

(9.4)

Introducing the non-dimensional parameter ε := δ/ℓex, we get

1

ε ℓex3 E(mδ; Ωδ) = Dε(m) :=

1

2

∫

Ω

|∇m|2 dτ + ε2(

1

2

∫

R3

|hd[m; Ω]|2 dτ +

∫

Ω

ψ(m) dτ

)

(9.5)

that we rewrite under the form

Dε(m) := D(m) + ε2F(m), with F(m) :=1

2

∫

R3

|hd[m; Ω]|2 dτ +

∫

Ω

ψ(m) dτ . (9.6)

For ε≫ 1, i.e. for samples much larger than the exchange length, the prominent terms in the energy are

the stray field energy and the anisotropy. Minimizing magnetization distributions are not uniform in these

situations because constant magnetizations induce large surface charges. The typical observed behavior is

in fact a partition of the sample into regions called domains where the magnetization is almost constant

separated by thin layers called domain walls of thickness comparable to ℓex where the energy concentrates.

Local Minimizers of GL

100

For fine particles ε≪1, it is expected that the cost of domain walls exceeds the cost of surface charges.

In this case, the magnetization is almost uniform inside the body and the particle is said to be single-

domain. At the limit, according to the Stoner-Wohlfarth theory [SW48] the magnetization is considered

as spatially uniform in the particle, that is, m∈U(Ω, S2) := u: Ω→ S2 : ∃σ ∈ S2 u≡σ a.e. in Ω. Inthis case, the micromagnetic energy of u≡ σ ∈U(Ω, S2) reduces to

Dε(u) = ε2|Ω|(

1

2σT ·Nσ+ ψ(σ)

)

(9.7)

where the effective demagnetizing tensor N is the 3× 3 matrix defined by,

Nσ=− 1

|Ω|

∫

Ω

hd[u] dτ . (9.8)

The tensor Neff inherits the properties of −hd[·, Ω] as a continuous orthogonal projector of L2(Ω, R3).

In particular Neff is a non negative symmetric matrix and its eigenvalues are bounded by 1. Also notice

that since m 7→−hd[m,Ω] is a linear pseudo-differential operator of order 0, the coefficients of Neff only

depend on the shape of Ω and not on its diameter.

Let us state a simple result supporting the Stoner-Wohlfarth approximation: in small particles, min-

imizers of the micromagnetic energy are almost constant.

Proposition 9.1. Let Ω be an open subset of R3 with a finite volume |Ω|, let ψ:S2→R be lower semi-

continuous and let ε> 0. Then if m is a global minimizer of Dε in H1(Ω, S2), it satisfies

‖∇m‖L22 ≤ |Ω|ε2. (9.9)

Proof. If m is a minimizer, then Dε(m)≤ ε2F(u) for any u≡σ∈U(Ω,S2). Choosing σ∈S2 minimizing

ψ, we have ‖∇m‖L22 ≤ 2ε2(F(u)−F(m))≤ ε2|Ω|σT ·N σ ≤ |Ω|ε2.

Moreover, A. De Simone established in [Sim95] that for ε > 0 small, the magnetization can not

substantially decrease its energy by moving away from the set of constant maps.

Proposition 9.2. (Corollary of Proposition 3.4. in [Sim95])

limε↓0

(

minm∈H1(Ω,S2)

1

ε2Dε(m)

)

= minu∈U(Ω,S2)

F(u). (9.10)

The Stoner-Wohlfarth approximation is almost never exact. Indeed, assume that u≡ σ ∈U(Ω, S2) is

a minimizer or even a critical point of Dε in H1(Ω, S2), the associated Euler-Lagrange equation at this

point reads,

−hd[u; Ω](x)+ Dψ(σ)∈ spanσ inΩ . (9.11)

Consequently, in the plane σ⊥, the components of hd[u,Ω] should be uniform in Ω. This turns out to be

wrong for general domains with the notable exception of Ω being a solid sphere or even a solid ellipsoid.

Indeed, in these latter cases, a well known result of potential theory (see [Kel10, Max73]), states that the

stray field induced by uniform magnetizations is also uniform inside Ω. For these special geometries the

effective stray field is point-wise related to u.


101

Proposition 9.3. (Maxwell [Max73]) If Ω is a solid ellipsoid then, the linear mapping u 7→−hd[u;Ω]|Ωmaps U(Ω, S2) into itself (i.e. −hd[u] =Nσ in Ω for u≡ σ).

The Fundamental Theorem for fine ferromagnetic particles of W.F. Brown is stated in this setting:

Theorem 9.4. (Brown [Bro68]: solid sphere case – Aharoni [Aha88]: prolate spheroids – [Fra11] (see also [AB09]):

general ellipsoids). Assume that Ω is a solid ellipsoid of unit diameter and that ψ is of class C1 on S2.

There exists εc> 0 such that for every ε∈ [0, εc), any minimizer of Dε in H1(Ω, S2) is uniform in Ω

In fact, in the above references, the result is established assuming that Aanis is a second order poly-

nomial, (ψ(σ) = ψ0 +ha · σ+ (1/2)σT ·Aσ). For this reason, we provide the reader with a general proof

of Theorem9.4 in Section 9.2.2.

Remark 9.1. The proof of Theorem 9.4 gives an explicit lower bounds for εc. We can also derive an

upper bound by a linear stability analysis of the uniform magnetizations. Unfortunately, these bounds

are not sharp (and do not match). The critical value εc is not known explicitly, but can be determined

by numerical means. This remark also applies to the constants introduced in our main result below.

Remark 9.2. When ψ is a second order polynomial function, the minimizers of Dε in U(Ω,S2) are easily

deduced from the coefficients of ψ and N. For example, if ψ≡0 and if Ω is the solid ellipsoid defined by

x∈R3: (x1/a1)

2 + (x2/a2)2 + (x3/a3)

2<r2 with semi-axes a1≥ a2≥ a3> 0, these minimizers are

• all the elements of U(Ω, S2) in the case a1 = a2 = a3 (sphere);

• the elements of the circle U(Ω, S2∩ spane1, e2) if a1 = a2>a3 (prolate ellipsoid);

• the two vectors ±e1 if a1>a2≥ a3 (elongated ellipsoid).

Proposition 9.1 and Brown’s theorem do not describe all the stable observable configurations in

ellipsoidal domains, since these results do not rule out the existence of non uniform local minimizers of

Dε. The main contribution of this paper consists in filling this gap, at least for smooth uniformly convex

particles.

Theorem 9.5. Let ψ∈C2(S2,R) and let Ω⊂R3 be a C2 uniformly convex domain with unit diameter.

Let ε> 0 and assume that m is a local minimizer of Dε in H1(Ω, S2), i.e., there exists η > 0 such that

for every u∈H1(Ω, S2),

‖u−m‖H1 6η =⇒ Dε(m)6Dε(u).

Then

i. there exist εF > 0 and CF ≥ 0 only depending on Ω and ψ such that:

ε< εF =⇒ ‖∇m‖L2(Ω) ≤ CF ε2.

ii. if moreover Ω is an ellipsoid, there exists εF

′ > 0 which only depends on Ω and ψ such that

ε< εF′ =⇒ m∈U(Ω, S2).


102

The uniformity in space of locally minimizing magnetizations in small ellipsoidal particles (stated in

the second part) was conjectured by Brown himself [Bro68].

The first part of the Theorem implies D(m)=O(ε4) for local minimizers of Dε in a smooth uniformly

convex particle. Thus the main contribution of the energy comes from the lower order term ε2F(m) which

leads to Dε(m) =O(ε2). This rules out the existence of high energy local minimizers, in particular, any

family (mε)ε<εF of local minimizers of (Dε)ε<εF converges up to extraction towards a critical point of

F in U(Ω, S2).

In the small particle limit ε= 0, we may believe that, for finding the observed magnetization distri-

butions, it is sufficient to replace the Dirichlet energy by the constraint m∈U(Ω, S2) and look for local

minimizers of F in this set. This is indeed the case when F admits only isolated local minimizer u in the

set of uniform magnetizations: in fact, it is known (cfr. theorem 4.3 in [Sim95]) that, for every isolated

local minimizer u ∈ U(Ω, S2) of F , there exists a family of magnetizations (mε)ε<ε0 such that mε is a

local minimizer of Dε and mε→u in L2 as ε↓0.On the other hand the situation is more complex when F admits a continuum of local minimizers in

U(Ω, S2), since in this case Dε may admit only finitely many local minimizers. This phenomenon called

configurational anisotropy is due to the slight deviation of mε from the set of uniform magnetizations

(see [CW98] and the rigorous analysis in [Sla10] for prism-shaped particles with D4 symmetry).

9.1.1 Locally minimizing p-harmonic maps.

In the proof of our main result, we establish a Liouville type result for harmonic maps that we believe

of independent interest. Since it does not make the proof more cumbersome, we state our result in the

setting of p-harmonic maps with values into a general Hilbert space.

Let Ω ⊂ Rd be a bounded open set, p > 1 and H be a Hilbert space. The p-Dirichlet energy of a

mapping m∈W 1,p(Ω, H) is defined as

E(m,Ω) :=1

p

∫

Ω

|∇m|Hp dτ , (9.12)

where | · |H stands for the norm in H . Given a closed subset S of H , we define W 1,p(Ω,S) to be the set

W 1,p(Ω,S)= m∈W 1,p(Ω, H) such that m(x)∈ S for almost every x∈Ω . (9.13)

We first address the question of whether local minimizers of E in W 1,p(Ω, S) are constant vector fields

(i.e.m≡σ∈S a.e. in Ω); in this case local minimizers would be global minimizers: If S were star-shaped

with respect to some point σ, it is pretty obvious that a local minimizer m of E in W 1,p(Ω, S) should

be a constant vector field (just compare E(m) to E((1 − ε)m +εσ) for ε↓0). In the general case, such

variations using convex combinations are not possible, and this is a classical difficulty for the study of the

regularity of harmonic maps . When S is a smooth manifold, critical points of E in W 1,p(Ω,S) are called

p-harmonic maps or simply harmonic maps in the case p=2. Such a map satisfies, at least formally, the

Euler Lagrange equations:

−∇ ·(

|∇m|Hp−2∇m)

∈ Tm(x)S a.e. inΩ. (9.14)


103

We prove the following result.

Theorem 9.6. Let Ω⊂Rd be a bounded open set, let p > 1 and let S be a closed subset of a Hilbert

space H . Assume that m is local minimizer of E in W 1,p(Ω,S). We have,

i. if p>d and Ω is star-shaped, then m is constant;

ii. if p= d and Ω is star-shaped, then ∇m is supported in λΩ for some λ< 1;

iii. if p<d and if Ω is a C2 uniformly convex domain then m is constant.

Remark 9.3. In case ii (p=d), if we knew thatm were analytic, we would be in a position to conclude,

using the unique continuation property, that m is uniform on Ω. Such a situation occurs when d= p=2

and S ⊂RN is an analytic embedded compact manifold. In this case a theorem of Morrey [Mor48] states

thatm is Hölder continuous. This allows us to localize in the target manifold, i.e. if U is a neighborhood

of m(x) in S, then there exists a neighborhood ω ⊂ Ω of x such that m(ω) ⊂ U . Using analytic local

charts ψ:U ⊂ S→R2, we see that ψ m is a critical point of a coercive functional of the form

∫

ω

∇uT ·A(u)∇u dτ

defined for u ∈ H1(ω, R2). The associated Euler-Lagrange equations now read as a non degenerate,

quasilinear elliptic system with analytic coefficients. The general regularity theory for these systems yields

the analyticity of ψ m. Hence m is analytic and thanks to i , spatially uniform.

Remark 9.4. In fact, we establish iii under a slightly weaker convexity assumption on Ω. The stated

assumption amounts to ask for the second fundamental form Ay on ∂Ω to be uniformly coercive, i.e. there

exists c>0 such that Ay(v,v)≥ c|v |2 for every y∈∂Ω and every v ∈Ty∂Ω. We can relax this hypothesis

by assuming that Ω⊂Rd is bounded, convex and of class C2 and that A(y) is coercive for almost every

y in ∂Ω.

There is a huge literature on the qualitative theory of harmonic maps dealing with existence, regularity

and singularity issues. We refer the reader to the review papers [EL78, EL88, EL83, EL95, Har97, Sim84]

and more recently [Hél02]. The regularity of p-harmonic maps has also been investigated, see e.g. [Fuc90,

Fuc93, HKL88] and [Luc88]. The proof of Theorem9.6 relies on ideas from the interior regularity theory

of minimizing harmonic maps, in particular we use the notion of inner variation and a refined version

of the monotonicity formula of Schoen and Uhlenbeck [SU82]. However the present paper is essentially

self-contained, mainly because we need a specific treatment of the boundary.

In section 9.2, we state a general stability result for perturbations Dε=D+ε2F of the Dirichlet energy:

Theorem 9.9. We show that the micromagnetic energy satisfies the relevant hypotheses and establish

that Theorem 9.5 follows from Theorem 9.9. We also prove Theorem 9.4 at the end of the section. We

establish Theorem 9.6 and Theorem 9.9 in sections 9.3 and 9.4, respectively. Eventually, in section 9.5

we discuss open questions and possible generalizations of our results.


104

9.1.2 Some useful result.

For a mapping m : Ω→H , we denote by 〈m〉∈H the mean value of m over Ω. In order to ease possible

further studies concerning the dependency of the constants appearing in Theorem 9.5 with respect to Ω

and ψ, we keep track of the constants in our estimates. In particular, we will use the Poincaré inequality

for functions with vanishing mean value. Let us recall that the value of the Poincaré constant admits a

universal bound in convex domains with prescribed diameter.

Proposition 9.7. (Poincaré inequality, see [Beb03], [PW60]) Let Ω⊂Rd be a bounded convex

open set. There exists CP ≥ 0 such that,

‖f −〈f 〉‖L2 ≤ CP ‖∇f ‖L2 for every f ∈H1(Ω). (9.15)

Moreover, the optimal constant satisfies CP ≤ δΩ/π where δΩ is the diameter of Ω.

We will also make use of the following variation of the Poincaré inequality when Ω is a smooth convex

domains, which contains the origin.

Proposition 9.8. (interior-boundary Poincaré inequality) Let Ω⊂Rd be a bounded smooth

convex domain, There exists CP′ ≥ 0 such that

(∫

Ω×∂Ω

|f(x)− f(y)|2 (n(y) · y)dx dHd−1(y)

)

1/2

≤CP′ Ω√

‖∇f ‖L2 for every f ∈H1(Ω), (9.16)

and CP′ ≤ 2

√(1+ (d+1)/π) δΩ, where δΩ is the diameter of Ω.

For the convenience of the reader, we establish (9.16) in Appendix A.

9.2 A general stability/rigidity result. Proof of Theorems 9.4 and 9.5

We obtain Theorem 9.5 as a particular case of the more general Theorem 9.9 given below which concerns

more general energies Dε:=D+ ε2F defined for functions m∈H1(Ω,S) where S is a closed subset of the

Hilbert space H .

Let us list the relevant hypotheses for this result. First, since our method relies on the tools developed

for the proof of Theorem 9.6.iii , we assume:

(H1). That Ω is a C2, uniformly convex domain of Rd with unit diameter and d≥ 3.

(H2). Next, we require that the functional F satisfies some regularity properties. Namely,

i. The functional F :L2(Ω,H)→R is differentiable. Denoting by k[p]∈L2(Ω,H) the gradient

of F at some point p∈L2(Ω,H) and by B the convex hull of S in H , we assume that there

exists C1≥ 0, such that

supp∈L2(Ω,B)

‖k[p]‖L2 ≤C1 .


105

ii. The mapping p ∈ L2(Ω, H) 7→ k[p] ∈ L2(Ω, H) is Gâteaux differentiable and there exists

C2≥ 0 such that

‖Dk[p] · q‖L2 ≤C2‖q‖L2 for every p∈L2(Ω,B), q ∈L2(Ω, H).

iii. If p∈L2(Ω, H) and if K is a compact subset of L2(Ω, H), the convergence

∥

∥

∥

k[p + tq]−k[p]

t−Dk[p] · q

∥

∥

∥

L2→→→→→→→→→→→→→→→→→→→→→→→→t↓0 0 ,

holds uniformly in q ∈K.

iv. If p∈H1(Ω,B), then k[p] belongs to H1(Ω, H) and there exists C3≥ 0 such that

‖∇k[p]‖L2 ≤ C3(1 +‖∇p‖L2) for every p∈H1(Ω,B).

For the exact rigidity result, we require:

(H3). The gradient k=∇L2F maps U(Ω, H) into itself. Moreover, there exists C3′ ≥ 0, such that

‖∇k[p]‖L2 ≤ C3′‖∇p‖L2 for every p∈H1(Ω, H). (9.17)

We also need S to be a smooth manifold with a large group of isometries:

(H4). The closed subset S is a smooth manifold. Moreover,

i. There exists a constant CS ≥ 0, such that for every σ ∈ S and every ζ ∈ TσS there exists

a smooth one parameter group R(t)t∈R of isometries of S such that R(0)σ = ζ and∥

∥R(0)∥

∥

∞≤CS|ζ |.

ii. There exists CS′ ≥ 0, such that

|(σ ′−σ) · ξ | ≤ CS′ |σ ′− σ |2|ξ | for every σ, σ ′∈S , ξ ∈NσS ,

where NσS denotes the orthogonal space to TσS in H .

Theorem 9.9. Let Ω⊂Rd, let S be a closed subset of some Hilbert H , let ε>0 and assume that m is

a local minimizer of Dε :=D+ ε2F in H1(Ω,S) then:

i. if hypotheses (H1-H2) hold, there exists CF ≥ 0 and εF > 0 such that

ε<εF =⇒ ‖∇m‖L22 ≤ CF ε

2.

ii. if moreover, (H3-H4) hold then, there exists εF′ such that if ε< εF

′ then m is constant in Ω.

This result is established in section 9.4.

Example 9.5. If Ω is bounded, Hypotheses (H2) are satisfied by functional F of the form

F(p)=F

[

1

2

∫

Ω×Ω

A(x1, x2 )(p(x1), p(x2))dx1dx2 +

∫

Ω

ψ(x, p)dx

]


106

with F ∈C2(R,R), A∈Cc2(Ω×Ω, S) where S is the space of continuous symmetric bilinear forms on H ,

A(x1, x2) is symmetric for almost every (x1, x2), and ψ ∈Cc2(Ω×H). If moreover ψ does not depend on

x and

∇x

∫

Ω

A(x, z)dz≡ 0 in Ω, for every σ ∈S ,

then F satisfies hypothesis (H3). This is precisely the situation arising in Theorem 9.5.ii (where F is

linear). Hypotheses (H3) and (H4) are satisfied for S = R, TN , SN−1, SON(R), S

N−1/−1, 1. More

generally, if H is a Hilbert space, these hypotheses are satisfied for example if: S =H ; S is the sphere

σ ∈ H ; |σ | = 1; S = Gσ where G is a manifold of O(H) which is also a subgroup and σ ∈ H \ 0;S =H/G when G is a discrete subgroup of O(H).

9.2.1 Proof of Theorem 9.5 (Regularity properties of the Micromagnetic Energy)

We assume here that p= 2, d= 3, S = S2 ⊂R

3 =H and that Ω ⊂R3 is a C2 uniformly convex domain

with unit diameter, i.e. that hypothesis (H1) holds. We fix ε > 0 and consider the perturbation of the

Dirichlet energy Dε= D+ ε2F introduced in (9.6). We assume that ψ ∈C2(S2,R) and that m is a local

minimizer of Dε in H1(Ω, S2).

Let us check step by step that the hypotheses of Theorem9.9 are satisfied. Writing

F =Fd +F loc with Fd(p) :=−1

2

∫

Ω

hd [p; Ω] · p dτ , F loc(p) :=

∫

Ω

ψ(p) dτ , (9.18)

we remark that F can be extended to a functional on L2(Ω,R3). Indeed, p 7→−hd[p; Ω] is a continuous

linear projector of L2(Ω,R3) while setting ψ(σ) := ρ(|σ |)ψ(σ/|σ |) where ρ ∈Cc∞(0, 2) satisfies ρ≡ 1 on

some neighborhood of 1 makes ψ ∈Cc2(R3) and F loc well defined on L2(Ω,R3).

The next propositions state that Fd and F loc comply to the requirements of Theorem9.9.

Proposition 9.10. The functional Fd satisfies hypothesis (H2).

Proof. i.-ii.-iii. Since p 7→−hd[p;Ω] is a linear projection on the closed subspace V of L2(Ω,R3), Fd is

a bilinear symmetric continuous (and nonnegative) functional on L2(Ω,R3), with norm bounded by 1/2.

Therefore it is infinitely continuously differentiable, the gradient of Fd at some point p∈L2(Ω,R3) being

given by kd[p]=−hd[p;Ω], while ∀q∈L2(Ω,R3) one has Dkd[p] · q=−hd[q;Ω]. We also have the bounds

‖kd[p]‖L2 ≤ ‖p‖L2, ‖Dkd[p] · q‖L2 ≤ ‖q‖L2. (9.19)

In particular, (H2)-i.-ii.-iii. hold with C1 = |Ω|√

and C2 = 1.

iv . We have to check that p 7→ kd[p] = −hd[p, Ω] maps H1(Ω, R3) into itself. For this, we invoke

Proposition 9.11 below and conclude that Fd satisfies (H2)-iv. with C3 =max(

1, |Ω|√ )

C3′′.

Proposition 9.11. ([CF01] Lemma 2.3.) Let Ω be a bounded domain of class C2. If p∈H1(Ω,R3),

then the restriction of hd[p,Ω] to Ω belongs to H1(Ω,R3). Moreover there exists a constant C3′′=C3

′′(Ω)

such that

‖hd[p]‖H1(Ω) ≤ C3′′ ‖p‖H1, for every p∈H1(Ω,R3).


107

Proposition 9.12. The functional F loc satisfies hypothesis (H2).

Proof. (H2)-i. For p∈L2(Ω,R3), since ψ ∈Cc2(R3), the gradient of F loc at p is given by

kloc[p](x) :=∇ψ(p(x)), ∀x∈Ω . (9.20)

We notice that the operator p∈L2(Ω,R3) 7→kloc [p]∈L2(Ω,R3) is continuous, with

‖kloc [p]‖L2 ≤ |Ω|√

‖∇ψ‖∞ . (9.21)

Thus F loc satisfies (H2)-i. with constant C1 = |Ω|√

‖∇ψ‖∞.

(H2)-ii. Now, since ψ∈C2, p∈L2(Ω,R3) 7→kloc[p]∈L2(Ω,R3), is Gâteaux differentiable and one has

Dkloc[p] · q= D2ψ(p) · q , for p, q ∈L2(Ω,R3). Therefore, F loc satisfies (H2)-ii. with C2 = ‖D2ψ‖∞.

(H2)-iii. Let p∈L2(Ω,R3) and let K be a compact set of L2(Ω,R3). For q∈K and x∈Ω and t>0,

we set

R(x; q , t) :=∇ψ(p(x) + tq(x))−∇ψ(p(x))

t− D2ψ(p(x)) · q(x),

and we remark that, using the Taylor-Lagrange formula, we have

R(x; q , t) = [D2ψ(p(x)+ tζ(x, t)q(x))−D2ψ(p(x))] · q(x) ,

where ζ(x, t) ∈ (0, 1). Let us introduce a parameter η > 0 and split Ω into Ωη := x : |q(x)|<η/t and

Ω \Ωη. We have the obvious bounds

∀x∈Ω \Ωη , |R(x; q , t)| ≤2‖D2ψ‖∞ |q(x)| ,

while

∀x∈Ωη , |R(x; q , t)| ≤ ω(η)|q(x)| ,

where ω is a modulus of continuity for D2ψ (recall that ψ is of class C2 and compactly supported). This

leads to∫

Ω

|R(x; q , t)|2dx ≤ ω2(η)‖q‖L22 +4‖D2ψ‖∞2

∫

Ω\Ωη

|q(x)|2dx. (9.22)

Since K is compact in L2(Ω,R3), K is bounded in L2(Ω,R3)

∃CK> 0, ∀q ∈K, ‖q‖L2≤CK ,

and the functions x∈Ω 7→ |q(x)|2 : q ∈K are uniformly equi-integrable. Therefore for η > 0 fixed the

integral in the right hand side of (9.22) goes to 0 as t↓0 uniformly in q ∈K. This leads to

limt↓0

supq∈K

∫

Ω

|R(x; q , t)|2dx

≤ CKω2(η).


108

Eventually, since η is arbitrary and ω(η)→ 0 as η↓0, the above limit vanishes, as required.

(H2)-iv . Let us now assume p∈H1(Ω,R3). We have to check that k[p] belongs to H1(Ω,R3). First,

since ψ∈Cc2(R3), the mapping x 7→kloc[p](x)=∇ψ(p(x)) belongs to H1(Ω,R3) and using the chain rule,

we have the estimate,

‖∇(kloc[p])‖L2 ≤ ‖D2ψ‖∞‖∇p‖L2. (9.23)

We deduce from (9.23) that F loc satisfies (H2)-iv. with C3 =‖D2ψ‖∞.

For the second part of Theorem 9.5, we invoke Proposition 9.3 that states that, in solid ellipsoids,

uniform magnetizations create uniform stray fields. Taking into account the (homogeneous) anisotropy,

we have the following result.

Proposition 9.13. If Ω is a solid ellipsoid, then (H3) holds, that is to say k=∇L2F maps U(Ω,R3)

into itself and moreover, there exists C3′ ≥ 0, such that

‖∇k[p]‖L2 ≤ C3′‖∇p‖L2 for every p∈H1(Ω,R3).

Proof. Let u≡ σ ∈U(Ω, S2), then

k[u](x) = −hd[u; Ω](x)+∇ψ(σ) for every x∈Ω.

By Proposition 9.3, we know that hd[u; Ω] is uniform inside Ω. Thus k[u]∈U(Ω, S2).

Let us now establish the estimate. With the notation of the proof of Propositions 9.10 and 9.12,

we have ∇k[p] = ∇(kloc[p]) − ∇(hd[p]), for every p ∈ H1(Ω, R3). For the first term, we have already

established the desired estimate in (9.23). Next, let p ∈ H1(Ω, R3). By linearity of p 7→ hd[p], we

have hd[p] = hd[〈p〉] + hd[p − 〈p〉]. Since Ω is a solid ellipsoid, it follows from Proposition 9.3 that

hd[〈p〉] is constant in Ω. Hence ∇(hd[p]) = ∇(hd[p − 〈p〉]). Using Proposition 9.11 and the Poincaré

inequality (9.15), we get

‖∇(hd[p])‖L22 ≤ (C3

′′)2(‖p−〈p〉‖L22 +‖∇p‖L2

2 ) ≤ (C3′′)2(1+CP

2 )‖∇p‖L22 .

This establishes the estimate of (H3) with C3′ =‖D2ψ‖∞+C3

′′ 1 +CP2

√

.

Eventually, we check that S2⊂R

3 satisfies (H4).

Lemma 9.14. The sphere S = S2 satisfies (H4).

Proof. i. Let σ ∈ S2, ζ ∈ σ⊥ = TσS

2 and call ζ ′ := σ × ζ. Let us define the one parameter group of

rotations R(t) = etA where A is the skew symmetric matrix given by Aσ ′ = ζ ′× σ ′. This group satisfies

R(0)σ=Aσ= ζ and the estimate∥

∥R(0)∥

∥

∞≤CS2|ζ | holds with CS2=1. ii. For σ∈S2, we have NσS =Rσ,

so we may assume ξ=λσ. We compute for σ ′∈ S2,

(σ− σ ′) · ξ=λ(1− σ ′ ·σ) =λ

2|σ ′− σ |2 .


109

Hence, (H4)-ii. holds with CS2′ = 1/2.

As a conclusion, by Propositions 9.10 and 9.12, Theorem 9.5.i is a consequence of Theorem 9.9.i and

then by Proposition 9.13 and Lemma 9.14, Theorem 9.5.ii follows from Theorem 9.9.ii.

9.2.2 Proof of Theorem 9.4

At this point, we have at hand all the tools to prove Theorem9.4.

Proof. (of Theorem9.4) Let Ω be an ellipsoid, assume that ψ is of class C2 and let m minimizing Dεin H1(Ω, S2). We denote by 〈m〉 the average value of m on Ω and define

σ=

〈m〉|〈m〉| if 〈m〉=/ 0 ,

any point in S2 otherwise.

Eventually, let u≡σ ∈U(Ω, S2). By optimality of m, we have, Dε(m) ≤ Dε(u)= ε2F(u), thus

‖∇m‖L22 ≤ 2ε2[F(u)−F(m)]. (9.24)

Next, by Proposition 9.10 and Proposition 9.12, F is differentiable in L2(Ω,R3) and its gradient is given

by k[p](x)=−hd[p; Ω](x)+∇ψ(p(x)). So,

F(u)−F(m) =−∫

0

1(∫

Ω

k[u+ t(m−u)] · (m−u) dτ

)

dt.

Since k[u] does not depend on x∈Ω, we obtain

F(u)−F(m)=−k[u] ·∫

Ω

m−u dτ +

∫

0

1(∫

Ω

(k[u]−k[u+ t(m−u)]) · (m−u) dτ

)

dt.

From the expression of k, this leads to

F(u)−F(m) ≤ |Ω|(1+‖∇ψ‖∞)〈m−u〉+1

2(1+‖D2ψ‖∞)‖m−u‖L2

2 .

We first bound the integral in the right hand side. By definition of u, 〈m−u〉= 〈m〉−σ=(|〈m〉|−1)σ.

On the other hand, since |m(x)|= 1 a.e. in Ω, we have 〈|m−〈m〉|2〉= 1− |〈m〉|2. Using |〈m〉| ≤ 1 and

Poincaré inequality, we obtain

|Ω| |〈m−u〉| = |Ω|(1−|〈m〉|) ≤ |Ω|(1− |〈m〉|2) ≤∫

Ω

|m−〈m〉|2 dτ ≤ CP2 ‖∇m‖L2

2 .

Next, we bound the last term ‖m−u‖L22 . Since m∈L2(Ω, S2) and u minimizes the distance |u− 〈m〉|

in S2, we have |u− 〈m〉| ≤ |m(x)− 〈m〉| for almost every x∈Ω which gives the bound ‖u− 〈m〉‖L2

2 ≤‖m−〈m〉‖L2

2 . Therefore

‖m−u‖L2 ≤ ‖m−〈m〉‖L2+‖〈m〉 −u‖L2 ≤ 2‖m−〈m〉‖L2 ≤ 2CP ‖∇m‖L2 (9.25)


110

from Poincaré inequality. Eventually, we have obtained,

F(u)−F(m) ≤ CP2 (3 +‖∇ψ‖∞+2‖D2ψ‖∞)‖∇m‖L2

2 .

Together with (9.24), we get that if ε2< 1/2CP2 (3 +‖∇ψ‖∞+2‖D2ψ‖∞), then m is constant.

9.3 Proof of Theorem 9.6

Throughout this section, we assume that Ω⊂Rd is a bounded open set which is star-shaped with respect

to some point x0∈Rd. Without loss of generality, we assume x0=0. We let p>1 and S be a closed subset

of some Hilbert space H . Eventually we assume that m∈W 1,p(Ω,S) is a local minimizer of E in this set

for theW 1,p-topology. In order to prove Theorem 9.6, we compare the energy ofm with some competitors

mtt∈(0,t0)⊂W 1,p(Ω,S). As already noticed, the usual perturbations of the form mt=m+ tϕ are not

allowed and we use instead the so-called inner variations

mt=m (idΩ + tϕ) (9.26)

for some suitable ϕ∈C∞(Ω,Rd). Notice that ϕ is not supposed to vanish on ∂Ω and hence we have the

following two possibilities.

• If (idΩ+ tϕ)(Ω)⊂Ω for t>0 small enough, idΩ+ tϕ is a diffeomorphism from Ω onto a open subset

of Ω, mt is well defined and mt∈W 1,p(Ω, H) since by the chain rule,

∂imt(x) = [∂im+ t∂iϕ(x) · ∇m](x+ tϕ(x)) for i= 1, ..., d and for a.e. x∈Ω. (9.27)

We also have mt(x)∈ S almost everywhere in Ω, so that mt∈W 1,p(Ω,S).

• If (idΩ + tϕ)(Ω)⊂Ω, then we first have to consider a extension m ∈W 1,p(O, S) defined on some

open neighborhood O of Ω and such that m|Ω =m. We then set mt=m (idΩ + tϕ).

In both cases, we havemt→→m as t↓0 in W 1,p(Ω,H) (we can see this by using the density of C∞(O,H)

in Wloc1,p(O, H)), and hence, by local optimality of m, there exists η= η(m , ϕ) such that

0<t< η =⇒ E(m) ≤ E(mt).

In what follows, we consider a family of such inner variations ((mtθ: =m (idΩ + tϕθ))t)θ∈J where

ϕθθ∈J ⊂C∞(Ω,Rd). We need the following Lemma.

Lemma 9.15. If ϕθθ∈J is compact in C1(Ω,Rd) then the convergence mtθ→→→→→→→→→→→→→→→→→→→→→→→→t↓0 m in W 1,p(Ω,H) is

uniform in θ ∈J .

Proof. Denoting by Lθ the Lipschitz constant of ϕθ, we set

η := infθ∈J

min

(

d(Ω,Oc)

‖ϕθ‖∞,

1

Lθ

)

> 0.

Then, as soon as 0<t< η, idΩ + tϕθ is a diffeomorphism of Ω onto a relatively compact subset of O and

mtθ is a well defined element of W 1,p(Ω,S) for every θ ∈ J .


111

Now, assume by contradiction, that there exist δ > 0 and two sequences (θk)k⊂J and (tk)k such that

tk↓0 and

‖mtkθk −m‖W 1,p>δ for every k≥ 0. (9.28)

Up to extraction, we may assume that (ϕθk)k converges towards ϕθ in C1(Ω, Rd), and therefore (id +

tkϕθk)k converges to id in C1(Ω,Rd). We thus see that

(∣

∣mtkθk∣

∣

p+∣

∣∇mtkθk∣

∣

p)

kis uniformly equi-integrable

in Ω. When H is a finite dimensional space, this is sufficient to conclude that(

mtk

θk)

is relatively compact

in W 1,p(Ω, H). We then have, up to extraction mtk

θk →→w ∈W 1,p(Ω, H). But we also have mtk

θk →→m in

D ′(Ω, H), so that we can identify w=m contradicting (9.28).

When H is not finite dimensional, we need another argument. First we notice that we can approximate

m in L1(Ω,H) by a sequence (sj)j of simple measurable functions (the range of sj is a finite set Aj⊂H).

We then set Hj := span ∪l≤jAl and call Pj the orthogonal projector on Hj. Since this projection is a

contraction, we have

|Pjm|(x)≤ |m|(x) and |∇Pjm|(x)≤ |∇m|(x) a.e. in Ω .

On the other hand Pjm(x)→m(x) almost everywhere in Ω, therefore by Lebesgue’s dominated conver-

gence Theorem, we have Pjm→m in W 1,p(Ω, H).

For any δ > 0, we fix j such that ‖Pjm − m‖W 1,p<δ/3. A direct computation using (9.27) and

the change of variable y = x + tkϕθk(x) shows that ‖Pjmtk

θk − mtkθk‖W 1,p ≤ (1 + O(tk))‖Pjm −

m‖W 1,p(

Ω+B(

0,tk‖ϕθk‖〈oo〉

)), so we also have ‖Pjmtkθk −mtk

θk‖W 1,p<δ/3 for k large enough.

Eventually, by the definition of Pj, we have Pjmtkθk = (Pjm) (id + tkϕ

θk) and from the finite

dimensional case, we know that for k large enough ‖Pjmtkθk −Pjm‖W 1,p<δ/3. These estimates yield the

desired contradiction

‖mtkθk −m‖W 1,p≤‖Pjmtk

θk −mtkθk‖W 1,p +‖Pjmtk

θk −Pjm‖W 1,p +‖Pjm−m‖W 1,p<δ

for k large enough.

In the sequel, we only use three kinds of inner variations. In section 9.3.1, we first consider dilations

with coefficient (1− t), which amounts to choose ϕ(x)=−x in (9.26). Since Ω is star-shaped with respect

to 0, there is no need to extendm outside Ω in that case. This turns out to be sufficient to establish parts

i and ii of Theorem 9.6. Then, assuming that Ω is C2 and convex, we introduce a particular extension

of m and consider dilations with coefficient (1 + t) which correspond to the choice ϕ(x) = x in (9.26).

In these two steps, we do not really need m to be a local minimizer, we only use the weaker first order

condition E(mt) ≥ E(m)+ o(t).

In Section 9.3.2, we consider translations of the domain, that is to say inner variations generated by

the family of perturbations ϕθ(x)= θθ∈Sd−1. For this step, we make use of the second order optimality

condition

E(mtθ) ≥ E(m)+ o(t2) .

Since Sd−1 is compact, by Lemma9.15, this optimality condition is satisfied uniformly in θ∈S

d−1 which

is required for our proof. This is the reason why we ask for m to be a local minimizer for the W 1,p-

topology (see the discussion in section 9.5).


112

9.3.1 Domain dilations (proof of parts i, ii of Theorem 9.6 and preliminaries for iii)

Proof. We introduce a first family of inner variations of m, namely for t∈ (0, 1) andx∈Ω,we set,

mt(x) := m((1− t)x).

Let us compare E(mt) and E(m). First, we compute ∇mt(x)= (1− t)∇m((1− t)x). This identity holds

in Lp(Ω), as soon as t∈ (0, 1) and by the change of variable y= (1− t)x, we get for t∈ (0, 1),

E(mt; Ω) = (1− t)p−dE(m; (1− t)Ω) .

By hypothesis, E(m; Ω) ≤ E(mt; Ω) for t small enough. Multiplying both sides of this inequality by

t−1(1− t)d−p and simplifying, we obtain

E(m; Ω \ (1− t)Ω)

t≤ 1− (1− t)d−p

tE(m; Ω) . (9.29)

Let us notice that if d< p, then the coefficient in the right hand side is negative and since the left hand

side is non negative, this leads to E(m; Ω)= 0 and so m is constant in Ω. This proves i.

When p= d, we get E(m; Ω \ (1 − t)Ω) = 0, which means that ∇m is supported in (1 − t)Ω. This

implies part ii of the Theorem.

From now on, we assume d > p and that Ω has Lipschitz regularity and is star-shaped with respect

to some non-empty open ball Bρ(0).

Lemma 9.16. (Zooming in) The trace u0 of m on ∂Ω belongs to W 1,p(∂Ω, S) and satisfies the

estimates,

1

p

∫

∂Ω

|∇u0|p(y)(y ·n(y))dHd−1(y) ≤ limt↓0

E(m;Mt−)

t≤ lim

t↓0

E(m;Mt−)

t≤ (d− p)E(m), (9.30)

with Mt− := Ω \ (1− t)Ω, for t∈ (0, 1).

Proof. The central inequality of (9.30) is obvious. Moreover, starting from (9.29) and taking the limsup

as t↓0, we get the last inequality of (9.30). We now study the left hand side of (9.29) and establish the

first inequality of (9.30). Let us first notice that by the trace Theorem (and since Ω is star-shaped with

respect to Bρ(0)), m admits a representative defined on Ω, still denoted by m, such that, if we set

ut(y)=m((1− t)y) for y ∈∂Ω and t∈ [0, 1), then t∈ [0, 1) 7→ut∈L1(∂Ω,S) is continuous. By definition,

the trace of m on ∂Ω is u0. Since S is closed, we have ut∈L1(∂Ω,S), for every t∈ [0, 1). In particular,

limt↓0ut = u0 in L1(∂Ω, H) . (9.31)

We now introduce the family of change of variables

ψt : ∂Ω× (0, 1)→Rd, (y, s) 7→ (1− st)y .


113

For t ∈ (0, 1), the map ψt defines a (bi-Lipschitz) diffeomorphism from ∂Ω × (0, 1) onto its image Mt−.

Thus, we can define a family vt∈H1(∂Ω× (0, 1),S) by vt(y, s) :=m(ψt(y, s)). Due to (9.31), we have

limt↓0vt=u0 in L1(∂Ω× (0, 1), H), (9.32)

with the abuse of notation u0(y, s)=u0(y).

Next, we show that ∇u0 ∈ Lp(∂Ω, Hd). By the chain rule, the gradient of vt with respect to any

tangential direction ξ ∈Ty∂Ω reads

(ξ · ∇y)vt(y, s) = (1− st)(ξ · ∇)m(ψt(y, s)).

We write ∇yvt(y, s)=(1−st)∇τm(ψt(y, s)), where, for x=ψt(y, s)∈ Ωt, ∇τm(x) denotes the projection

of ∇m(x) onto Ty∂Ω×H ⊂Rd×H . The change of variable formula leads to

1

tp

∫

∂Ω×(0,1)

|∇yvt|p(y, s) (1− st)−pJt(y, s)dsdHd−1(y) =

1

tp

∫

Mt−|∇τm|p ≤ E(m;Mt

−)

t

where Jt denotes the Jacobian determinant of ψt. Using the orthogonal decomposition of Rd as Ty∂Ω⊕Rn(y)≃Ty∂Ω⊕R, we compute,

Dψt(y, s) =

(

(1− ts)idTy∂Ω −t(y− (y ·n(y))n(y))

0 −t(y ·n(y))

)

.

Hence, Jt(y, s)= t(1− ts)d−1(y ·n(y))= [1 +O(t)]t(y ·n(y)) and the above identity simplifies to

[1 +O(t)]1

p

∫

∂Ω×(0,1)

|∇yvt|p(y, s) (y ·n(y))dsdHd−1(y) =1

tp

∫

Mt−|∇τm|p ≤ E(m;Mt

−)

t. (9.33)

Since Ω is star-shaped with respect to Bρ(0) the weight y · n(y) is uniformly bounded from below by

a positive constant on ∂Ω. On the other hand, we know from (9.29) that the right hand side of (9.33)

remains bounded as t↓0. Hence, the family ∇yvtt∈(0,1/2) is bounded in Lp(∂Ω× (0, 1)) and therefore,

up to extraction, ∇yvt weakly converges in Lp(∂Ω × (0, 1)). We already know from (9.32) that (vt)tconverges towards u0 in D ′(∂Ω× (0, 1), H), and we can thus identify the limit and deduce:

∇yvt →→→→→→→→→→→→→→→→→→→→→→→→t↓0 ∇yu0, weakly in Lp(∂Ω× (0, 1)) . (9.34)

Consequently, u0∈W 1,p(∂Ω, S) as claimed. Eventually, by lower semi-continuity of the Lp-norm under

weak convergence, we get the first inequality of (9.30) by sending t↓0 in (9.33).

We now establish that the inequalities (9.30) are in fact identities.

Lemma 9.17. (Zooming out) The following identities hold.

1

p

∫

∂Ω

|∇u0(y)|p(y ·n(y)) dHd−1(y) = limt↓0

E(m;Mt−)

t= (d− p)E(m) . (9.35)


114

Proof. Thanks to the higher regularity of u0 = m|∂Ω established in Lemma 16, we are able to use

extensions of m that are only built on u0. Let s⋆ > 0 be small enough such that ψ: (y, s) ∈ ∂Ω × (0,

s⋆) 7→ y+ sn(y)∈Rd defines a (bi-Lipschitz) diffeomorphism onto its image N . We set O := Ω∪N and

define the extensionm∈W 1,p(O,S) of m by

m(x) =

m(x) if x∈Ω,

u0(y) if x= ψ(y, s).(9.36)

Let t⋆>0 be such that (1+ t⋆)Ω⊂O, we now define a new inner perturbation ofm bymt(x) :=m((1+ t)x)

for x∈Ω, t∈ (0, t⋆).

By the local optimality of m, we know that E(m) ≤ E(mt) for t > 0 small enough. Then, using the

homogeneity of the energy and the splitting of (1 + t)Ω into Ω∪Mt+, with Mt

+ := (1+ t)Ω \Ω, we get

E(m) ≤ E(mt) = (1 + t)p−dE(m; Ω) + (1+ t)p−dE(m;Mt+).

Multiplying by (1+ t)d−p leads to

1

p

∫

Mt+|∇m|p(z)dz ≥ [(1 + t)d−p− 1]E(m).

Dividing by t and letting t↓0, we obtain

liminft↓0

1

tp

∫

Mt+|∇m|p(z)dz ≥ (d− p)E(m). (9.37)

But, on Mt+,m is given in terms of u0 which enables us to express the left hand side as a function of

∇u0. Using the change of variable z= (1 + ts)y (as in the previous Lemma), we compute

1

tp

∫

Mt+|∇m|p(z)dz =

1

p

∫

0

1[∫

∂Ω

|∇m|p((1 + ts)y)(n(y) · y)dHd−1(y)

]

(1+ ts)d−1ds

Next, from the identitym(y+ sn(y)) =u0(y) and the chain rule, we have for every ξ ∈Ty∂Ω,

ξ · ∇u0(y) = ([ξ+ s(ξ · ∇)n(y)] · ∇)m(y+ sn(y)),

and thus

∇m(y+ sn(y)) = [1+O(s)]∇u0(y) .

Denoting by π the orthogonal projection on ∂Ω, this yields

1

tp

∫

Mt+|∇m|p(z) dz = [1+O(t)]

1

p

∫

∂Ω×(0,1)

|∇u0|p(π[(1 + ts)y])(n(y) · y)dHd−1(y)ds.

Now, we let t↓0. Since the family of mappings (y, s) 7→ π[(1 + ts)y] converges to (y, s) 7→ y in

C1(∂Ω× (0, 1)) as t↓0 and since y 7→ |∇u0|p(y)(n(y) · y) belongs to L1(∂Ω), we get,

limt↓0

1

tp

∫

Ωt

|∇m|p(z)dz =1

p

∫

∂Ω

|∇u0|p(y)(n(y) · y)dHd−1(y) .


115

Together with (9.37), this leads to

1

p

∫

∂Ω

|∇u0|p(y)(n(y) · y)dHd−1(y) ≥ (d− p)E(m).

Using this inequality and (9.30), we get (9.35).

The identities of Lemma 9.17 allow us to convert the weak convergence of Lemma 9.16 to strong

convergence. Indeed:

Lemma 9.18. We have the strong convergence

[∇m] ψt →→→→→→→→→→→→→→→→→→→→→→→→t↓0 ∇u0 in Lp(∂Ω× (0, 1)),

where we recall the notation ψt (y, s): =(1− st)y for y ∈ ∂Ω and s, t∈ (0, 1).

This Lemma plays the role of a regularity result form near the boundary. It will be crucial for carrying

out the computations in the next sections.

Proof. Using the change of variable z= ψt(y, s)= (1− st)y as in the proof of Lemma 16, we get

1

p‖[∇m] ψt‖L⋆

pp = [1 +O(t)]

E(m;Mt−)

t.

where we have denoted by ‖v‖L⋆p the norm in Lp(∂Ω× (0, 1)) defined by

‖v‖L⋆p :=

(

∫

∂Ω×(0,1)

|v |p(y, s) (y ·n(y))dHd−1(y) ds

)

1

p

.

By Lemma 9.17, we know that

limt↓0

‖[∇m] ψt‖L⋆p = ‖∇u0‖L⋆

p < ∞.

Thus, there exists a w ∈Lp(∂Ω× (0, 1)) and a sequence tk↓0 such that

(

[∇m] ψtk

)

k∈N→ w weakly in Lp(∂Ω× (0, 1)). (9.38)

Moreover, by the lower semi-continuity of the Lp-norm under weak convergence

‖w‖L⋆p 6 ‖∇u0‖L⋆

p . (9.39)

We now claim that

w = ∇u0 (9.40)

Assuming the claim, we get from (9.38) and (9.39)

(

[∇m] ψtk

)

k∈N−→ ∇u0 strongly in Lp(∂Ω× (0, 1)).


116

Eventually, since the limit ∇u0 does not depend on the particular subsequence (tk), we deduce that the

whole family [∇m] ψt converges towards ∇u0 as t↓0. This establishes the Lemma, assuming (9.40).

Let us now establish (9.40). Using the notation ∂nm(ψt(y, s)) :=n(y) · [∇m](ψt(y, s)) (notice that

this extends the classical normal derivative) and orthogonal decompositions, we have

[∇m] ψt = [∇τm] ψt + n⊗ ∂nm ψt, w = wτ + (w ·n)n.

With this decomposition, (9.38) reads

[∇τm] ψtk → wτ , ∂nm ψtk → w ·n both weakly in Lp(∂Ω× (0, 1)) as k↑∞.

Now, in the proof of Lemma9.16 we have seen that [∇τm]ψt weakly converges to∇u0 in Lp(∂Ω× (0,1)),

and therefore wτ =∇u0. Taking into account (9.40) we end with

‖w‖L⋆p 6 ‖∇u0‖L⋆

p = ‖wτ‖L⋆p. (9.41)

Since wτ(y, s) and n(y) are orthogonal in H , this yields w=wτ =∇u0 and establishes the claim.

Remark 9.6. The preceding Lemma implies the following strong form of the Neumann boundary con-

ditions:

[∂nm] ψt →→→→→→→→→→→→→→→→→→→→→→→→t↓0 0 in Lp(∂Ω× (0, 1)).

9.3.2 Domain translations (proof of theorem 9.6.iii)

For the time being, we have considered inner variations produced by dilations of the domain with respect

to x0 = 0. When Ω is the unit ball, these variations do not rule out non-constant mappings of the form

m(x) =m0(x/|x|). Indeed, for p < d and m0 ∈W 1,p(Sd−1, S) such mappings do belong to W 1,p(Ω, S)

and are homogeneous of degree 0, (mt=m). Moreover, the identities (9.35) of Lemma 9.17 remain true

regardless of whether m is a local minimizer. In such cases our previous computations are not sufficient

and looking for second order optimality conditions would not improve the situation if we stick on the same

variations. We therefore consider below different inner variations, namely those produced by translations

of the domain. From now on, we assume that the domain Ω is (still) bounded, convex and of class C2.

We also assume that 0 ∈ Ω, which implies that Ω is still star-shaped with respect to some non-empty

open ball Bρ(0) and the results of the previous section 9.3.1 apply. We also recall that 1< p<d, and in

particular, d≥ 2.

We introduce a new extension of m. Let t⋆> 0 be such that

φ : (y, s)∈ ∂Ω× (−t⋆, t⋆) 7−→ y+ sn(y)∈Rd

defines a bi-Lipschitz diffeomorphism onto its range. For t∈ (0, t⋆), we set,

N t+ := φ(∂Ω× (0, t)) and N t

− := φ(∂Ω× (−t, 0)).


117

We extend m on x∈R3 : d(x,Ω)< t⋆= Ω∪N t⋆

+ by setting

m(x) : =

m(x) if x∈Ω,

m(φ(y,−s)) if x= φ(y, s), (y, s)∈ ∂Ω× [0, t⋆).(9.42)

Since φ is a bi-Lipschitz diffeomorphism, we havem∈W 1,p(Ω,S). Next, for θ∈Sd−1, and t∈ (0, t⋆), we set

mtθ(x) := m(x+tθ) for every x∈Ω.

It is clear that mtθ ∈W 1,p(Ω,S) and by Lemma 15, there exists t′∈ (0, t⋆] such that

E(m) ≤ E(mtθ) for every t∈ (0, t′) and θ ∈ S

d−1.

Averaging in θ ∈Sd−1, we set for t∈ (0, t′),

Q(t) :=1

t2

∫

Sd−1

[E(mtθ)−E(m)]dHd−1(θ) ≥ 0 for t∈ (0, t′). (9.43)

The main task of this Section is to establish the following lemma.

Lemma 9.19. For ξ ∈ Ty∂Ω, let us denote by ay(ξ) = Ay(ξ, ξ) the quadratic form associated to the

second fundamental form Ay of ∂Ω at y. We have

limt↓0

Q(t) = − |Sd−1|2d

∫

∂Ω

ay(∇u0(y))|∇u0|p−2(y)dHd−1(y). (9.44)

The factor ay(∇u0(y)) in the integrand of (9.44) deserves somme comment. Since Ω is of class C2, ayis well defined as a quadratic form on Ty∂Ω. As usual, we extend this quadratic form to Rd by setting

ay(ξ) := ay

(

πy ξ)

where πy denotes the orthogonal projection onto Ty∂Ω. Next, given, any Hilbertian

basis B of H , we can extend the domain ay to continuous linear forms V : Rd→H . Indeed, writing the

decomposition of V into the Hilbertian basis as V (z)=∑

b∈B Vb(z)b, for z ∈Rd, we set

ay(V ) :=∑

b∈B

ay(Vb).

With this convention, the integrand in the right hand side of (9.44) is well defined with

ay(∇u0(y)) =∑

b∈B

ay(∇(u0(y) · b)) =∑

b∈B

Ay(∇(u0(y) · b),∇(u0(y) · b)).

We postpone the proof of Lemma 9.19, and show first that it implies Theorem 9.6.iii .

Proof. (of Theorem 9.6, iii) We assume that Ω is uniformly convex or, at least, that ay is coercive

for almost every y ∈ ∂Ω (see Remark 9.4). Assuming that (9.44) holds, with (9.43), this leads to

∫

∂Ω

ay(∇u0)|∇u0|p−2dHd−1(y) ≤ 0 .


118

Since ay is coercive almost everywhere on ∂Ω, we see that ∇u0 vanishes in Lp(∂Ω). The identities of

Lemma 9.17 then lead to E(m)=0 and m is constant, as claimed.

Proof. (of Lemma 9.19) To lighten notation, we set q(z) :=1

p|∇m |p(z) for z ∈Ω∪N t⋆. By Fubini,

we rewrite Q(t) as

Q(t) :=1

t2

∫

Rd

wt(z)q(z)dz,

with wt(z) = Hd−1(θ ∈ Sd−1 : z ∈ Ω + tθ) − Hd−1(Sd−1 )1Ω(z). Obviously, if z ∈ Ω is such that

d(z, ∂Ω)≥ t, then wt(z)= 0. For z ∈ ∂Ω +Bt(0), we distinguish the cases z ∈N t+ and z ∈N t

−.

wt(z) =

Hd−1(θ ∈Sd−1 : z ∈Ω + tθ) if z ∈N t+,

−Hd−1(θ ∈Sd−1 : z ∈ [Rd \Ω] + tθ) if z ∈N t

−,

0 if z ∈Rd \ [N t

+∪N t−].

(9.45)

Using the change of variables z= φ(y, rt)= y+ rtn(y), we obtain

Q(t) =1

t

∫

∂Ω

∫

−1

1

wt φ(y, rt) q φ(y, rt)Jφ(y, rt) drdHd−1(y),

where Jφ denotes the Jacobian determinant of φ. We rewrite this expression under the form

Q(t) =1

t

∫

∂Ω

∫

0

1

[(wtφ) (qφ) Jφ](y, rt)− [(wtφ) (qφ )Jφ](y,−rt)drdHd−1(y). (9.46)

In order to obtain the limit of this integral as t↓0, we compute the first order expansions in t of wt, q

and Jφ.

•First, using the orthogonal decomposition Rd=Ty∂Ω⊕Rn(y)≃Ty∂Ω⊕R, we compute

Dφ(y, s) =

(

idTy∂Ω +sDn(y) 0

0 1

)

which gives, uniformly in y ∈ ∂Ω and r ∈ (−1,1),

Jφ(y, rt) = 1+ rtκ(y) + o(t), (9.47)

where κ(y)=TrAy denotes the total curvature of ∂Ω at y.

• Next, for r∈ (0, 1), using m(φ(y, rt)) =m(φ(y,−rt)), we compute

Dm(φ(y, rt)) = Dm(φ(y,−rt)) ·Dφ(y,−rt) ·Dφ−1(y, rt)

= Dm(φ(y,−rt)) · [id− 2rtDn(y) + o(t)].


119

This yields the expansion, using the shorter notation y±= y± rtn(y)

q(y+) = q(y−)− 2rt(DmT (y−) ·Dn(y) ·Dm(y−))|∇m |p−2(y−)+ o(t)q(y−)

that we rewrite as

q(y+) = q(y−)− 2rt ay(∇m(y−))|∇m|p−2(y−)+ o(t)q(y−), (9.48)

uniformly in y ∈ ∂Ω, r ∈ (0,1).

• Eventually, we establish that

wt(φ(y,±rt))|Sd−2| = ±Θ0(r)− tΘ1(r)κ(y) t+ o(t), (9.49)

holds uniformly in y ∈ ∂Ω, r ∈ (0,1) and with the notation

Θ0(r) : =

∫

0

acos r

(sin ϕ)d−2dϕ, Θ1(r) :=(1− r2)

d−1

2

2(d− 1).

Proof. (of (9.49)) Let y ∈ ∂Ω. Without loss of generality, we use local coordinates for which

y= 0, n(y) = ed and we identify Ty∂Ω with Rd−1. By definition, for t small enough,

wt(φ(y,+rt)) = Hd−1(θ ∈Sd−1 : rt ed+tθ ∈Ω),

wt(φ(y,−rt)) = −Hd−1(θ ∈ Sd−1 :−rt ed+ tθ ∈R

d \Ω). (9.50)

Since Ω is of class C2, using a local chart, we can parameterize locally ∂Ω as the graph of a C2

concave function. Calling Dρ the (d− 1)-ball Dρ= Ty∂Ω∩Bρ(y)⊂Rd−1, there exists ρ> 0 and

hy∈C2(Dρ,R) such that

∂Ω∩ [Dρ× (−ρ, ρ)] = (ξ, hy(ξ)) ; ξ ∈Dρ.

With our hypotheses, hy satisfies

hy(0)= 0, ∇hy(0) =0, and D2hy(0)(ξ, η) =−Ay(ξ, η) for every ξ, η ∈Rd−1 =Ty∂Ω.

Moreover (∂Ω being compact) there exists CΩ≥ 0 that only depends on Ω such that uniformly in

y ∈ ∂Ω

‖D2hy‖∞ ≤ CΩ .

Notice also that ρ only depends on Ω.

We now estimate (9.50). Let ξ ∈ Sd−2⊂R

d−1 and let us study the intersection of θ ∈ Sd−1 ;

−r t ed + tθ ∈ Rd \ Ω with the half-plane Red ⊕ R+ξ. For this, we consider the following

parametrization of the semi-circle Sd−1∩ [R ed⊕R+ξ],

eϕ := (cos ϕ) ed+(sin ϕ)ξ , for ϕ∈ [0, π].


120

ed

y ξϕξ,r,t

Ω

tSd−1\Ω

eϕξ,r,t

Figure 9.1. For ξ ∈ Sd−2 ⊂Rd−1 we study the intersection of θ ∈ Sd−1 ;−rt ed + tθ ∈Rd \Ω with thehalf-plane Red ⊕R+ξ.

For r ∈ (0, 1), −rt ed+ teϕ∈Rd \Ω if and only if

fξ,r,t(ϕ) := − t cos ϕ+ rt+hy(t (sin ϕ) ξ) < 0.

Performing the Taylor expansion of fξ,r,t(ϕ) at t= 0 leads to

fξ,r,t(ϕ)

t= − cos ϕ+ r+

t sin2ϕ

2D2hy(0)(ξ, ξ) + o(t) .

Now, noticing that fξ,r,t(0)/t=(r− 1)< 0 and fξ,r,t(π)/t=(1− r)> 0, we see that fξ,r,t changes

sign only once for t sufficiently small. Consequently, fξ,r,t(ϕ)<0 on [0, ϕξ,r,t) where ϕξ,r,t∈ (0, π)

is the unique solution in (0, π) of fξ,r,t(ϕ) = 0 (see Figure 9.1). The Taylor expansion of ϕξ,r,twrites

ϕξ,r,t = acos(r)− t

2D2hy(0)(ξ, ξ)(sin acos(r))+ o(t).

Therefore, integrating in ξ ∈Sd−2, we get

wt(φ(y,−rt)) = −∫

Sd−2

∫

0

ϕξ,r,t

(sin ϕ)d−2dϕ

dHd−2(ξ)

= −Hd−2(Sd−2)

∫

0

acos r(sin ϕ)d−2dϕ

+t

2

(∫

Sd−2

D2hy(0)(ξ, ξ)dHd−2(ξ)

)

(1− r2)d−1

2 + o(t).

By the use of the identity∫

Sd−2(e · ξ)2dHd−2(ξ) = |Sd−2|/(d − 1) for all e ∈ S

d−2 and the

diagonalization of D2hy(0) in an orthonormal basis, we have

∫

Sd−2

D2hy(0)(ξ, ξ)dHd−2(ξ) =Hd−2(Sd−2)

d− 1TrD2hy(0) = − Hd−2(Sd−2)

d− 1κ(y).


121

Substituting this identity in the expression above, we obtain the expected expansion of wt(φ(y,

−rt)). The computation for wt(φ(y, rt)) is similar and can be obtained by substituting −hy for

hy. This establishes (9.49).

We are now able to compute the limit of Q(t) as t↓0. Plugging the expansions (9.47), (9.48) and (9.49)

in (9.46), we get:

Q(t) = − |Sd−2|∫

∂Ω

κ(y)

∫

0

1

(Θ1(r)− rΘ0(r))q(φ(y,−rt))drdHd−1(y)

+

∫

∂Ω

∫

0

1

2rΘ0(r) ay(∇m(φ(y,−rt)))|∇m|p−2(y)drdHd−1(y)

+o(1)

∫

∂Ω

∫

0

1

q(φ(y,−rt))drdHd−1(y)

.

(9.51)

To justify the passage to the limit t↓0, we prove the following property which is a direct consequence of

Lemma 9.18 and of the convexity of the domain.

Lemma 9.20. Let Ψt0≤t be the family of functions defined on ∂Ω× (0, 1) by

Ψt(y, r) = y− rtn(y) = φ(y,−rt).

Then [∇m] Ψt →→→→→→→→→→→→→→→→→→→→→→→→t↓0 ∇u0 in Lp(∂Ω× (0, 1)).

Proof. Let λ :=1/min |y |; y∈∂Ω>0. We use the change of variable (y, r)=Ψt−1 ψλt(y, s), where we

recall that ψλt (y, s)= (1− sλt)y for y ∈ ∂Ω, s, t∈ (0, 1). We have

∫

∂Ω×(0,1)

|[∇m] Ψt(y, r)−∇u0(y)|pdHd−1(y)dr

=

∫

ψλt−1(

Ωt

)

|[∇m] ψλt(y, s)−∇u0(y)|pJt(y, s)dHd−1(y)ds.

where Jt denotes the Jacobian determinant of Ψt−1 ψλt. We easily check by direct computation that Jt

is uniformly bounded. By Lemma 18, the right hand side integral goes to 0 as t↓0.

Passing to the limit t↓0 in (9.51) we get (using Lemma 9.20 in the three integrals)

limt↓0

Q(t) = − |Sd−2|(∫

0

1

Θ1(r)− rΘ0(r)dr

)∫

∂Ω

H(y)|∇u0|p(y)

pdHd−1(y)

+

(∫

0

1

2rΘ0(r)dr

)∫

∂Ω

ay(∇u0(y))|∇u0|p−2(y)dHd−1(y)

.


122

Let us now compute the integrals in r. Using Fubini and then integrating by parts, we get:

∫

0

1

rΘ0(r)dr =

∫

0

1 ∫

0

acos r(sin ϕ)d−2dϕdr

=

∫

0

π/2

(sin ϕ)d−2

(∫

0

cos ϕ

rdr

)

dϕ

=1

2(d− 1)

∫

0

π/2

(sin ϕ)ddϕ,

while, using the change of variable r= cos ϕ,

∫

0

1

Θ1(r)dr =1

2(d− 1)

∫

0

π/2

(sin ϕ)ddϕ =

∫

0

1

rΘ0(r)dr.

Calling Wd=∫

0

π/2(sin ϕ)ddϕ the Wallis integrals we get

limt↓0

Q(t) = − |Sd−2|Wd

d− 1

∫

∂Ω

ay(∇u0(y))|∇u0|p−2(y)dHd−1(y).

This expression can be further simplified using that Wallis integrals satisfy the classical relation Wd/(d−1) =Wd−2/d and |Sd−1| = 2|Sd−2|Wd−2. We therefore have |Sd−2|Wd/(d − 1) = |Sd−1|/(2d), leading to

(9.44). This completes the proof of Lemma 9.19 and thus of Theorem 9.6.


Let Ω ⊂Rd, d ≥ 3, let S be a closed subset of some Hilbert space H and F : L2(Ω, H) →R. The proof

of Theorem 9.9 proceeds as follows. In section 9.4.1, considering Dε as a perturbation of E (in the case

p=2), we go through the same steps as for the unperturbed case to obtain an inequality of the form

D(m) ≤ ε2[L(m)+L′(m) +Q(m)],

where, roughly speaking, L(m), L′(m) and Q′(m) are respectively linear in k[m]⊗∇m, ∇[k[m]]⊗∇mand Dk[m]⊗∇m. The estimates of hypothesis (H2) then lead to

|L(m)+L′(m)| ≤ κL‖∇m‖L2 (1 + ‖∇m‖L2), |Q(m)| ≤κQ‖∇m‖L22 .

Simplifying, we obtain ‖∇m‖L2≤ 2κL

1− 2(κQ + κL)ε2ε2 which proves Theorem 9.9.i.

Eventually, in Section 9.4.2, we assume that (H3-H4) hold and we compare Dε(m) with Dε(Rtm)

when Rt is a continuous group of isometries of S. Since the Dirichlet energy is invariant by isometry

of the target we deduce that 〈m〉 is almost a critical point of F . This fact and hypothesis (H3) then

lead to the quadratic estimate |L(m)+L′(m)|≤κQ′ ‖∇m‖L22 instead of the linear one we had before. As

a consequence we have ‖∇m‖L22 ≤2(κQ+ κQ

′ )ε2‖∇m‖L22 . Hence, for ε< 1/ 2(κQ+κQ

′ )√

, m is constant.

9.4.1 Proof of Theorem 9.9.i (inner variations)

In this subsection, we assume that Hypotheses (H1) and (H2) hold. Let us first state the counterpart

of Lemma 9.16.


123

Lemma 9.21. The trace u0 of m on ∂Ω belongs to H1(∂Ω,S), with the estimate

1

2

∫

∂Ω

|∇u0(y)|p(y ·n(y))dH2(y)+ ε2∫

Ω

k[m](x) · (x · ∇m(x))dx ≤ D(m). (9.52)

Proof. We proceed as in the proofs of Lemma 9.16, use the same notation and skip the details. By local

optimality of m, Dε(m) ≤Dε(mt) for t > 0 small enough, where mt(x) =m((1 − t)x) for t ∈ [0, 1) and

x∈Ω. Since

Dε(mt)= (1− t)−1D(m; (1− t)Ω)+ ε2F(mt),

proceeding as in Lemma 9.16„ we are led to

D(m;Mt−)

t≤ D(m)+ε2(1− t)

F(mt)−F(m)

t, (9.53)

(with Mt−= Ω \ (1− t)Ω). Using the differentiability of F , we rewrite the last term as

F(mt)−F(m)

t=

(

k[m],mt−mt

)

L2

+ o(∥

∥

∥

mt−mt

∥

∥

∥

L2

)

. (9.54)

Now, since m∈H1(Ω, H), we have for almost every x∈Ω,

mt−mt

(x)=−∫

0

1

x · ∇m((1− ts)x)ds=−x · ∇m(x)−∫

0

1

x · [∇m((1− ts)x)−∇m(x)]ds

Squaring, integrating on Ω and then using Jensen inequality and Fubini, we get

∫

Ω

∣

∣

∣

∣

mt(x)−m(x)

t+x · ∇m(x)

∣

∣

∣

∣

2

≤∫

Ω

|x|2∫

0

1

|∇m((1− ts)x)−∇m(x)|2dsdx.

≤∫

0

1(∫

Ω

|∇m((1− ts)x)−∇m(x)|2dx)

ds.

Since∇m∈L2(Ω,Hd), the family of maps vλ1/2<λ<1 defined by vλ(x) :=∇m(λx) is relatively compact

in L2(Ω) (proceed as in the proof of Lemma 9.15). We already know that vλ→→→→→→→→→→→→→→→→→→→→→→→→→→→ →λ↑1 ∇m in the sense of

distributions, so this convergence also holds in L2(Ω). In particular, the last integral goes to 0 as t↓0 and

we conclude that (1/t)[mt−m]→→→→→→→→→→→→→→→→→→→→→→→→t↓0 (x 7→−x · ∇m(x)) in L2(Ω).

Coming back to (9.54), we get

F(mt)−F(m)

t=−

∫

Ω

k[m](x) · (x · ∇m(x))dx+ o(1).

In particular, the right hand side of (9.53) is bounded as t↓0 and we have,

limt↓0

D(m;Mt−)

t≤ D(m)−ε2

∫

Ω

k[m](x) · (x · ∇m(x))dx.


124

We then conclude as in the proof of Lemma 9.16.

Next, we revisit the proof of Lemma 9.17.

Lemma 9.22. The following identities hold.

1

2

∫

∂Ω

|∇u0(y)|2(y ·n(y))dH2(y)+ ε2∫

Ω

k[m](x) · (x · ∇m(x))dx

= limt↓0

D(m;Mt−)

t= D(m).

(9.55)

Moreover m satisfies the conclusions of Lemmas 9.18 and 9.20 with p=2.

Proof. We proceed as in the proof of Lemma 9.17, we extendm on a neighborhood O⊃Ω by the function

m defined by (9.36). We then set mt(x)=m((1+ t)x) for x∈Ω and t∈ (0, t⋆). The optimality of m now

leads to

D(m) ≤ D(m; (1 + t)Ω \Ω)

t+ ε2(1+ t)

F(mt)−F(m)

t.

Sincem∈H1(Ω,O), we obtain, as in the previous proof,

F(mt)−F(m)

t=

∫

Ω

k[m](x) · (x · ∇m(x))dx+ o(1).

The proof of (9.55) is now a copy of that of Lemma 9.17. Lemmas 9.18 and 9.20 follow since their proof

only rely on the last equality of (9.55) and on the convexity of Ω.

Eventually we consider translations of the domain as in section 9.3.2.

Lemma 9.23. We have

∫

∂Ω

ay(∇u0(y)) dH2(y) ≤ ε2∑

i=1

d ∫

Ω

Dk[m] · ∂im · ∂im− ∂ik[m] · ∂im dτ . (9.56)

Proof. We use the notation of section 9.3.2, and define

Q(t) : =1

t2

∫

Sd−1

[D(mtθ)−D(m)]dHd−1(θ), R(t) : =

1

t2

∫

Sd−1

[F(mtθ)−F(m)]dHd−1(θ) (9.57)

and Qε: =Q+ ε2R.

By local optimality of m, we know that for t > 0 small enough Qε(t) ≥ 0. The computations of

section 9.3.2 leading to (9.44) remain valid in the present context. In particular, by Lemma 9.22 we can

use Lemma 9.20. Consequently,

limt↓0

Q(t) = − |Sd−1|2

∫

∂Ω

ay(∇u0(y))dHd−1(y). (9.58)


125

Let us now compute the limit of R(t). Since F is continuously differentiable, we have for t > 0 small

enough and every θ ∈Sd−1,

F(mtθ)−F(m)

t=

∫

0

1 ∫

Ω

k[mstθ ](x) · (θ · ∇mst

θ (x))dxds.

Rewriting the integrand as

k[mstθ ] · (θ · ∇mst

θ ) = k[mstθ ] · (θ · ∇m) + k[mst

θ ] · (θ · ∇mstθ −m),

and integrating in θ ∈Sd−1, we obtain,

R(t) =1

2t

∫

Sd−1

∫

0

1 ∫

Ω

k[mstθ ]−k

[

mst−θ]

(x) · (θ · ∇m)(x)dxdsdHd−1(θ)

+1

t

∫

Sd−1

∫

0

1 ∫

Ω

k[mstθ ](x) · (θ · ∇mst

θ −m)(x)dxdsdHd−1(θ)

=: R1(t)+R2(t).

For the first term, we notice that m ∈H1(Ω + Bt⋆(0)) implies that ξ ∈Rd 7→m(ξ + ·) ∈ L2(Ω,Rd)

is differentiable in Bt⋆(0) with differential ∇m(ξ + ·). Since ∇m ∈ L2(Ω + Bt⋆(0)), we deduce that

(m(ξ+ ·)−m)/|ξ |ξ∈Bt⋆(0) is relatively compact subset of L2(Ω,H). The differentiability properties of

p∈L2(Ω,R3) 7→k[p]∈L2(Ω,R3) stated in Hypotheses (H2)ii-iii then yield

k[m(ξ+ ·)]−k[m] = Dk[m] · (ξ · ∇m) + o(|ξ |)in L2(Ω, H). This leads to

k[mstθ ]−k

[

mst−θ]

t− 2sDk[m] ·(θ · ∇m) →→→→→→→→→→→→→→→→→→→→→→→→t↓0 0 in L2(Ω, H) uniformly in (θ, s)∈ S

d−1× (0, 1).

Integrating in (θ, s)∈Sd−1× (0, 1), we obtain

R1(t) →→→→→→→→→→→→→→→→→→→→→→→→t↓0 1

2

∫

Sd−1

∫

Ω

Dk[m] ·(θ · ∇m)(x) · (θ · ∇m)(x)dxdHd−1(θ).

And since∫

S2 (θ⊗ θ)dH2(θ)= (|Sd−1|/d)id, we get

R1(t) →→→→→→→→→→→→→→→→→→→→→→→→t↓0 |Sd−1|2d

∫

Ω

∑

i=1

d

Dk[m] ·(∂im) · ∂im. (9.59)

Next, to evaluate the term R2(t), we first integrate by parts to get

R2(t) = −∫

Sd−1

∫

0

1 ∫

Ω

(θ · ∇k[mstθ ])(x) · mst

θ −m(x)t

dxdsdHd−1(θ)

+

∫

Sd−1

∫

0

1 ∫

∂Ω

k[mstθ ](y) · mst

θ −m(y)t

(θ ·n(y))dH2(y)dsdHd−1(θ)

=: R2,1(t) +R2,2(t).


126

The expansion mstθ −m= stθ · ∇m+ o(t) in L2(Ω, H) being valid uniformly in (θ, s)∈ S

d−1× (0, 1) we

obtain by Hypothesis (H2).iv :

R2,1(t) →→→→→→→→→→→→→→→→→→→→→→→→t↓0 − 1

2

∫

Sd−1

∫

Ω

(θ · ∇k[m]) · (θ · ∇m)dHd−1(θ)=−|Sd−1|2d

∫

Ω

∑

i=1

d

∂ik[m]∂im. (9.60)

We now establish that the boundary term R2,2(t) goes to 0 as t↓0.Then (9.56) will follow from (9.58),

(9.59), (9.60) and the local optimality of m. Writing mstθ −m=st

∫

0

1θ · ∇mrst dr , we get

R2,2(t) =

∫

Sd−1

∫

0

1

s

∫

∂Ω

∫

0

1

k[mstθ ] · (θ · ∇mrst

θ )(θ ·n)drdHd−1dsdHd−1(θ).

Using Fubini and the change of variables r= q/s, we compute,

R2,2(t) =

∫

Sd−1

∫

∂Ω

[

∫

(0,1)2k[mst

θ ] · (θ · ∇mqtθ )(θ ·n) dqds

]

dHd−1dHd−1(θ).

By Hypothesis (H2).iv and the trace Theorem, the mapping ξ ∈ Bt⋆(0) 7→ k[p]|∂Ω ∈ L2(∂Ω, H) is

continuous. In particular,

(y, s) 7→k[mstθ ](y) →→→→→→→→→→→→→→→→→→→→→→→→t↓0 k[m] in L2(∂Ω× (0, 1)) uniformly in θ ∈S

d−1.

On the other hand, by Lemma 9.20, we also have

(y, q) 7→ (θ · ∇mqtθ )(y) →→→→→→→→→→→→→→→→→→→→→→→→t↓0 k[m] in L2(∂Ω× (0, 1)) uniformly in θ ∈S

d−1.

Consequently, R2,2(t) converges, as t↓0, towards∫

∂Ω

k[m] ·(∫

Sd−1

(θ · ∇u0)(θ ·n)dH2(θ)

)

dH2 =|Sd−1|d

∫

∂Ω

k[m] · (n · ∇u0)dH2 = 0.

For the last identity, we recall that u0 is defined as an element of H1(∂Ω,R3), so n ·∇u0≡0 on ∂Ω. This

ends the proof of the Lemma.

We are now able to establish the first part of Theorem 9.9. By Hypothesis (H1), Ω is uniformly

convex and 0∈Ω, so there exists cΩ> 0 such that

cΩay(ξ) ≥ (y ·n(y))|ξ |2, for every y ∈ ∂Ω, ξ ∈Ty∂Ω. (9.61)

Lemmas 9.22 and 9.23 then lead to

D(m) ≤ ε2 L(m) +L′(m)+Q(m). (9.62)

with

L(m): =

∫

Ω

k[m](x) · (x · ∇m) (x)dx

Q(m) := cΩ

∫

Ω

∑

i=1

3

(Dk[m] · ∂im · ∂im), L′(m) :=−cΩ∫

Ω

∑

(∂ik[m] · ∂im).


127

Using the bounds of Hypothesis (H2) we have:

L(m) ≤ C1‖∇m‖L2, L′(m) ≤ cΩC3‖∇m‖L2(1 + ‖∇m‖L2), Q(m) ≤ cΩC2‖∇m‖L22 . (9.63)

Plugging these estimates in (9.62), and simplifying, we obtain

‖∇m‖L2 ≤ CF ε2 for ε< εF, (9.64)

with

εF :=1

2 cΩ(C2 +C3)√ , CF := 4(C1 +C3).

This establishes Theorem 9.9.i .

9.4.2 Proof of Theorem 9.9.ii (target variations)

Let us now assume that hypotheses (H3-4) also hold. We show that in this case, the right hand side

of (9.62) is bounded by Cε2‖∇m‖L22 . We already have the desired quadratic estimate for Q(t) (last

inequality of (9.63)). Next, by Hypothesis (H3) we also have

|L′(m)| ≤ cΩC3′‖∇m‖L2

2 . (9.65)

The more difficult part is be to establish that there exist εL> 0 and CL≥ 0 depending on Ω and F such

that for 0<ε<εl, we have:

|L(m)| ≤ CL‖∇m‖L22 , (9.66)

Taking this estimate for granted, we end the proof as follows. Using (9.65), (9.66) and the last estimate

of (9.63) to bound the right hand side of (9.62), we get, for ε< εL,

(1− 2(cΩ(C2 +C3′)+CL)ε2)D(m) ≤ 0.

Thus m is constant as soon as ε<εF′ :=min

(

εL, 1/ 2(cΩ(C2 +C3′) +CL)

√)

.

To end the proof we have to establish (9.66). Let us first write L(m)=L1(m)+L2(m) with

L1(m): =k[〈m〉] ·∫

Ω

(x · ∇m) dx, L2(m): =

∫

Ω

k[m]−k[〈m〉] · (x · ∇m) dx, (9.67)

where, with an abuse of notation, we identify k[m] with its constant value N · 〈m〉+∇ψ(〈m〉)∈H inside

Ω.

For the second term, we use the Cauchy-Schwarz inequality and the differentiability of p ∈L2(Ω, H) 7→k[p]∈L2(Ω, H) with the estimate of Hypothesis (H2)ii to get

|L2(m)| ≤ C2‖m−〈m〉‖L2‖∇m‖L2 ≤ C2CP ‖∇m‖L22 . (9.68)

Let us now bound L1(m). Let σ ∈H be a projection of 〈m〉 on S, i.e. σ ∈ argmin|σ ′−〈m〉|2 ; σ ′∈S.First, by definition of σ (with σ ′ =m(x)∈S) we have:

|σ−〈m〉|2 =1

|Ω|

∫

Ω

|σ−〈m〉|2 ≤ 1

|Ω|

∫

Ω

|m(x)−〈m〉|2dx ≤ CP2

|Ω|‖∇m‖L22 .


128

By triangular inequality this leads to

‖σ−m‖L2 ≤ 2CP ‖∇m‖L2 (9.69)

Now, we integrate by parts to obtain

L1(m) = d|Ω|k[〈m〉] · (〈m〉∂ −〈m〉), with 〈m〉∂: = 1

d|Ω|

∫

∂Ω

m(y)(y ·n(y))dHd−1(y).

Let us perform the orthogonal decomposition:

〈m〉∂ −〈m〉 =: ξm + ζm, with ξm∈NσS , ζm∈TσS.

Let ξ ∈NσS such that ξ · ξm = |ξm| and |ξ |=1. Using hypothesis (H4) and (9.69) we have

|(〈m〉− σ) · ξ | =

∣

∣

∣

∣

1

|Ω|

∫

Ω

[m(x)−σ] · ξdx∣

∣

∣

∣

≤ CS′

|Ω|

∫

Ω

|m(x)−σ |2dx ≤ 4CS′CP

2

|Ω| ‖∇m‖L22

We have a similar estimate for |〈m〉∂ − σ | which leads to,

|ξm| ≤ 2CS′(

6CP2 +CP

′ 2)

|Ω| ‖∇m‖L22 .

We conclude that

|d|Ω|k[〈m〉] · ξm| ≤ 2dC1CS′(

6CP2 +CP

′ 2)‖∇m‖L22 . (9.70)

We now bound the term d|Ω|k[〈m〉] · ζm. First, we have an obvious linear control of ζm:

|ζm| ≤∣

∣〈m〉∂−〈m〉∣

∣ ≤ CP′

d|Ω|√ ‖∇m‖L2. (9.71)

Eventually, we use the optimality of m to establish: that the following estimate holds

|Ω||k[〈m〉] · ζ | ≤ (2C1CS +C2)CP ‖∇m‖L2|ζ | for every ζ ∈TσS. (9.72)

Let ζ ∈TσS. By hypothesis (H4) there exists a smooth one parameter group of isometries of S, R(t)t∈R,

such that R(0) ·σ= ζ and∥

∥R(0)∥

∥

∞≤CS|ζ |.Let us set γt :=R(t) ·m, for t∈R. By local optimality of m the function f(t) :=F(γt) admits a local

minima at t=0. In particular,

0 = f ′(0) = DF(m) ·

R(0) ·m

=

∫

Ω

k[m] ·(

R(0) ·m(x))

dx. (9.73)

Now, let us write

|Ω|k[〈m〉] · ζ =

∫

Ω

k[〈m〉] · ζ dτ

=

∫

Ω

k[〈m〉]−k[m] · ζ dτ +

∫

Ω

k[m] ·

R(0) · (σ−m)

dτ +

∫

Ω

k[m] · R(0) ·m dτ


129

By (9.73), the last term vanishes. Using the Cauchy Schwarz inequality and (9.69), the second term

satisfies the estimate∣

∣

∣

∣

∫

Ω

k[m] ·

R(0) · (σ−m)

dτ

∣

∣

∣

∣

≤ 2C1CSCP ‖∇m‖L2 |ζ |,

For the first term, we have:

∣

∣

∣

∣

∫

Ω

k[〈m〉]−k[m] · ζ dτ

∣

∣

∣

∣

≤ C2‖〈m〉−m‖L2|ζ | ≤ C2CP ‖∇m‖L2|ζ |.

The last two inequalities imply (9.72) which together with (9.70) and (9.71) yield |L1(m)| ≤C‖∇m‖L22

for some C ≥ 0 depending on d, |Ω|, C1, C2, CS′ , and CS. This ends the proof of Theorem 9.9.

9.5 Concluding remarks and further generalizations.

Let us discuss how our results depend on the shape of the domain Ω. Paying attention to the constants

in the estimates, we see that the parameters CF, εF and εF′ in Theorem 9.9. only depend on cΩ, the

Poincaré constant CP , CP′ and the constants C1, C2 and C3 of Hypothesis (H2). The Poincaré constant

are uniformly bounded since Ω is a convex domain with unit diameter.

In the context of micromagnetism (Theorem 9.5) the constants C1 and C2 only depend on ψ, but C3

also depends on the constant C3′ of Proposition 9.11 for which we do not have an explicit bound. It would

be interesting to know whether this constant admits a uniform bound in the set of smooth convex domains.

The constant cΩ (introduced in (9.61)) is the inverse of the minimal curvature of ∂Ω. In particular,

this constant blows up when considering a sequence of unit diameter convex domains (Ωk)k such that Ωkis included in the thin cylinder BR2(0, 1)× (−1/k, 1/k). This includes the case of thin ellipsoids. So, our

result degenerates in the limit of thin ellipsoids.

We do not claim that the uniform convexity assumption on the domain is sharp. However, we believe

that the results do not hold in some complex geometries. For example, if Ω⊂R3 is the ball with cavity,

Ω = x∈R3 ; 1< |x|< 2

then we believe that the non-constant mapping m(x) :=x/|x| is a local minimizer of D in H1(Ω, S2). In

the perturbed case, we may conjecture that for ε small enough, we can find in the neighborhood of the

set R ·m : R∈ SO3(R) some local minimizers of Dε in H1(Ω, S2).

In the proof of Theorem 9.6, we only test the local optimality of m under a small set of variations:

small dilations of the domain and (for which we only need a first order optimality condition) and small

translations of the domain. The nature of the target set S does not play any role in the proof.

In the proof of Theorem 9.9.ii , we also use the optimality of m with respect to the target set. If we

denote by mtt∈(−t′,t′) ⊂H1(Ω, S) the trajectory corresponding to one of these variations, we require

that Dε(m)≤Dε(mt) for |t|<t′ small enough. In the case of domain translations, we also used that this

property holds uniformly in every direction of translation. We do this when integrating the optimality

condition on the set of directions θ∈Sd−1 in (9.43). We could avoid this if we already knew thatm were

smooth, in this case we would prove the counterpart of (9.44) without integrating in θ. In this situation,

we could weaken the optimality hypothesis to

Dε(m) ≤ limt↓0

Dε(mt), ∀mt∈C1([0, t′), L2(Ω, S2)), m0 =m andd

dtm|t=0 =/ 0. (9.74)


130

We did not succeed in providing a proof with this weaker assumption in the general case. Let us mention

however that when Ω is the unit open ball centered at 0, then n(y) = y on ∂Ω and we can use the

expansions m of m in place of the expansion m (see their definitions in (9.36), (9.42)). In this case,

the proof simplifies: we only need to consider domain translations in the directions e1, e2, e3, so that

Theorem 9.9 holds under the weaker assumption (9.74).

Theorems 9.6 and 9.9 may be generalized. First, if M is an invertible matrix of Rd×d, we see that,

by the change of variable z=Mx, Theorem 9.6 holds for the functional

EM(m) :=1

p

∫

|M · ∇m|p.

We may also consider small perturbations of the form,

E η(m) :=1

p

∫

Ω

|∇m|p + η

∫

Ω

a(x,∇m)

with η > 0, and a∈C1(Ω×Hd,R), such that ∇xa∈C1(Ω×Hd,R) and

sup(x,v)Ω×H

∣

∣Dxa∣

∣(x, v)+∣

∣Dx

2 a∣

∣(x,v)

|v |p +|Dva(x,v)|

|v |p−1 <∞.

We obtain, with reasonable modifications of the current proof that Theorem 9.6 holds for Eη under the

condition η < ηc where ηc> 0 depends on cΩ and a.

9.6 Appendix A (proof of Proposition 9.8)

Let Ω⊂Rd be a bounded convex smooth open set with diameter δ>0 and assume that 0∈∂Ω. We consider

a real valued function f ∈C∞(Ω), (the result for f ∈H1(Ω) is obtained by density of C∞(Ω) in H1(Ω)

and by continuity of the trace mapping f ∈H1(Ω) 7→ f|∂Ω∈L2(∂Ω)). We have to estimate the quantity

I(f) :=

∫

Ω

∫

∂Ω

|f(x)− f(y)|2(n(y) · y)dHd−1(y)dx.

For y ∈ ∂Ω, we define te following weighted mean value of f along the segment (0, 1)y:

〈f 〉y: =d+ 1

2

∫

0

1

rd−1

2 f(ry)dr.

We then decompose f(y) as 〈f 〉y+ [f(y)−〈f 〉y] to get I(f)≤ 2 (|Ω| I1(f) + I2(f)), with

I1(f) :=

∫

∂Ω

|f(y)−〈f 〉y |2(n(y) · y)dHd−1(y),

I2(f) :=

∫

Ω

∫

∂Ω

|f(x)−〈f 〉y |2(n(y) · y)dHd−1(y)dx.

9.6 Appendix A (proof of Proposition 9.8)

131

We start by estimating I1(f). Let us fix y ∈ ∂Ω, we have,

f(y)−〈f 〉y =(d+ 1)

2

∫

0

1

rd−1

2 (f(y)− f(ry))dr

=(d+ 1)

2

∫

0

1

rd−1

2 (1− r)

∫

0

1

y · ∇f((r+ (1− r)s)y)dsdr

Using the change of variable s= (t− r)/(1− r) and then Fubini, we obtain,

f(y)−〈f 〉y=(d+ 1)

2

∫

0

1

rd−1

2

∫

r

1

y · ∇f(ty)dtdr=

∫

0

1

y · ∇f(ty) td+1

2 dt.

Squaring and using the Jensen inequality, we get:

|f(y)−〈f 〉y |2 ≤ |y |2∫

0

1

|∇f(ty)|2 td+1 dt ≤ δ2∫

0

1

|∇f(ty)|2 td−1 dt.

Then, we multiply by (y ·n(y)) and integrate in y ∈ ∂Ω. Using the change of variable z = ψ(y, t) := ty,

which maps ∂Ω× (0, 1) onto Ω \ 0, we get

I1(f) ≤ δ2∫

Ω

|∇f |2(z) [t(z)]d−1 (y(z) ·n(y(z)))

Jψ(ψ−1(z))dz = δ2

∫

Ω

|∇f |2(z)dz, (9.75)

with the notation, ψ−1(z) =: (y(z), t(z)) and Jψ(y, t) = detDψT ·Dψ√

(y, t). Indeed, introducing the

orthogonal decomposition Rd = Ty∂Ω ⊕ Rn(y) ≃ Ty∂Ω ⊕ R, we compute the Jacobian matrix of ψ in

these spaces:

Dψ(y, t) =

(

t idTy∂Ω (y− (y ·n(y))n(y))

0 (y ·n(y))

)

.

The Jacobian determinant of ψ is Jψ(y, s)= td−1(y ·n(y)).

Now we bound I2(f). We first use the definition of 〈f 〉y and the Cauchy-Schwarz inequality to get

for every (x, y)∈Ω× ∂Ω:

|f(x)−〈f 〉y |2 ≤ (d+ 1)2

4

∫

0

1

|f(x)− f(ry)|2 rd−1dr

Integrating in y∈ ∂Ω, and using the change of variable z= ψ(y, r) as above, we obtain (after integration

in x∈Ω):

I2(f) ≤ (d+1)2

4

∫

Ω×Ω

|f(x)− f(z)|2dxdz ≤ (d+ 1)2|Ω|CP2 ‖∇f ‖L22 . (9.76)

Inequality (9.16) follows from (9.75) and (9.76) with

CP′

δ= 2

[

1+ (d+1)2(

CPδ

)

2]

√

≤ 2√ (

1+(d+1)CP

δ

)

.

Since CP/δ ≤ 1/π, we have CP′ /δ≤ 2

√(1+ (d+ 1)/π) as claimed.


132

10Composite Ferromagnetic Materials

This chapter is devoted to the presentation of the results obtained in collaboration with Prof. François

Alouges during my second PhD internship at CMAP, Ecole Polytechnique, Palaiseau (Paris).

The purpose of this chapter is to rigorously derive the homogenized functional of a periodic mixture

of ferromagnetic materials. We thus describe the Γ-limit of the Gibbs-Landau free energy functional,

as the period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to 0.

10.1 Introduction

Composite materials are an important class of natural or engineered heterogeneous media, composed of

a mixture of two or more constituents with significantly different physical or chemical properties, firmly

bonded together, which remain separate and distinct within the finished structure. Finding a model which

considers the composite as a bulk and whose coefficients and terms are computed from suitable averages

of those of its constituents and the geometry of the microstructure is the aim of homogenization theory.

The study of composites and their homogenization is a subject with a long history, which has attracted

the interest and the efforts of some of the most illustrious names in science [MP00]: In 1824, Poisson, in

his first Mémoire sur la théorie du magnétisme [Poi24], put the basis of the theory of induced magnetism

assuming a model in which the body is composed of conducting spheres embedded in a nonconducting

material. This paper is the origin of the basic models and ideas that prevailed in the theory of hetero-

geneous media in almost all domains of continuum mechanics, for almost a century after its appearance

[MP00]. We refer the reader to [MP00] and [Lan78] for more historical details.

133

Nowadays, non-homogeneous and periodic ferromagnetic materials are the subject of a growing

interest. Actually such periodic configurations often combine the attributes of the constituent materials,

while sometimes, their properties can be strikingly different from the properties the different constituents

[Mil02]. These periodic configurations can be therefore used to achieve physical and chemical proper-

ties difficult to achieve with homogeneous materials. To predict the magnetic behavior of these composite

materials is of prime importance for applications [Mil02].

From a mathematical point of view, the study of composite materials, and more generally of media

which involve microstructures, is the main source of inspiration for the Mathematical Theory of

Homogenization which, roughly speaking, is a mathematical procedure which aims at understanding

heterogeneous materials with highly oscillating heterogeneities (at the microscopic level) via a homoge-

neous model [Nan07].

The main objective of this paper is to perform, in the framework of De Giorgi’s notion of Γ-

convergence [DF75] and Allaire’s notion of two-scale convergence [All92] (see also the paper by

Nguetseng [Ngu89]), a mathematical homogenization study of the Gibbs-Landau free energy func-

tional associated to a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which

the heterogeneities are periodically distributed inside the ferromagnetic media. Compared to earlier works

(see [Sim93, Sim95] for instance) our present contribution considers mixtures of different materials and

the full Gibbs-Landau functional.

10.1.1 The Landau-Lifshitz micromagnetic theory of single-crystal ferromagnetic materials

According to Landau-Lifshitz-Brownmicromagnetic theory of ferromagnetic media (see [LL35, Bro63,

BMS09]), the states of a rigid single-crystal ferromagnet, occupying a region Ω ⊆R3, and subject to a

given external magnetic field ha, are described by a vector field, the magnetization M, verifying the so-

called fundamental constraint of micromagnetic theory : a ferromagnetic body is always locally saturated,

i.e. there exists a positive constant Ms such that

|M|=Ms(T ) a.e. in Ω. (10.1)

The saturation magnetization Ms depends on the specific material and on the temperature T that

vanishes above the Curie point. Since we will assume that the specimen is at a fixed temperature below

the Curie point of the material, the value Ms will be regarded as a material dependent function, and

therefore as a constant function when working on single-crystal ferromagnets. Due to the constraint

(10.1) in the sequel we express the magnetization M under the form M :=Ms(T )m where m: Ω→ S2 is

a vector field which takes its values on the unit sphere S2 of R3.

Even though the magnitude of the magnetization vector is constant in space, in general it is not the

case for its direction, and the observable states can be mathematically characterized as local minimizers

of the Gibbs-Landau free energy functional associated to the single-crystal ferromagnetic particle:

GL(m) :=

∫

Ω

aex|∇m|2 dτ +

∫

Ω

ϕan(m) dτ − µ0

2

∫

Ω

hd[Msm] ·Msm dτ −µ0

∫

Ω

ha ·Msm dτ .

=: E(m) =:A(m) =:W(m) =:Z(m)

(10.2)

Composite Ferromagnetic Materials

134

The first term, E(m), is called exchange energy, and penalizes spatial variations of m. The factor

aex in the term is a phenomenological positive material constant which summarizes the effect of (usually

very) short-range exchange interactions. The second term, A(m), or the anisotropy energy, models the

existence of preferred directions for the magnetization (the so-called easy axes). The anisotropy energy

density ϕan: S2→R

+ is assumed to be a non-negative, even, and globally lipschitz continuous function,

that vanishes only on a finite set of unit vectors, the easy axes, and is a function that depends on the

crystallographic symmetry of the sample. The third term, W(m), is the magnetostatic self-energy,

and is the energy due to the (dipolar) magnetic field, also known in literature as the stray field, hd[m]

generated by m. From the mathematical point of view, assuming Ω to be open, bounded and with a

Lipschitz boundary, a given magnetization m∈L2(Ω,R3) generates the stray field hd[m] =∇um where

the potential um solves:

∆um =−div (mχΩ) in D ′(R3). (10.3)

In (10.3) we have indicated with mχΩ the extension of m to R3 that vanishes outside Ω. Lax-Milgram

theorem guarantees that (10.3) possesses a unique solution in the Beppo-Levi space [DLAC00]:

BL(R3) =

u∈D ′(R3) :u(·)

1 + | · | ∈L2(R3) and ∇u∈L2(R3,R3)

. (10.4)

Eventually, the fourth term Z(m), is called the Zeeman energy, and models the tendency of a specimen

to have its magnetization aligned with the external field ha, assumed to be unaffected by variations of m.

The competition of those four terms explain most of the striking pictures of the magnetization that

ones can see in most ferromagnetic material [HS08], in particular the so-called domain structure, that

is large regions of uniform or slowly varying magnetization (the magnetic domains) separated by very

thin transition layers (the domain walls).

10.1.2 The Gibbs-Landau energy functional associate to composite ferromagnetic materials

Physically speaking, when considering a ferromagnetic body composed of several magnetic materials (i.e.

a non single-crystal ferromagnet) a new mathematical model has to be introduced. In fact, as far as the

ferromagnet is no more a single crystal, the material functions aex,Ms(T ) and ϕan are no longer constant

on the region Ω occupied by the ferromagnet. Moreover one has to describe the local interactions of two

grains with different magnetic properties at their touching interface [AFM06].

From a mathematical point of view, this latter requirement is usually taken into account in two

different ways. Either one adds to the model a surface energy term which penalizes jumps of the mag-

netization direction m at the interface of both grains, or, and we stick on this later on, one simply

considers a strong coupling, meaning that the direction of the magnetization does not jump through an

interface. We insist on the fact that only the direction is continuous at an interface while the magnitude

Ms isobviouslydiscontinuous. The natural mathematical setting for the problem becomes to consider that

the magnetization directionm∈H1(Ω,S2) the Sobolev metric space, that for technical reasons, we endow

with the induces strong L2(Ω) metric. It is in this framework that we will conduct our work from now on.

10.1 Introduction

135

Figure 10.1. If we assume that the heterogeneities are evenly distributed inside the ferromagnetic media Ω, wecan model the material as periodic. As illustrated in the figure, this means that we can think of the material asbeing built up of small identical cubes Qε, the side length of which we call ε.

We start by recalling the basic idea of the mathematical theory of homogenization. Let Ω ⊂ R3

be the region occupied by the composite material. If we assume that the heterogeneities are regularly

distributed, we can model the material as periodic. As illustrated in Figure 10.1, this means that we

can think of the material as being built up of small identical cubes, the side length of which being

called ε. Let Q = R3/Z3 ≃ [0, 1)3 be the periodic unit cube of R3. We let for y ∈ Q, aex(y), Ms(y),

ϕan(y,m) be the periodic repetitions of the functions that describe how the exchange constant aex, the

saturation magnetization Ms and the anisotropy density energy ϕan(y,m) vary over the representative

cell Q (see Figure 10.1). Substituting x/ε for y, we obtain the «two-scale» functions aε(x) := aex(x/ε),

Mε(x) :=Ms(x/ε) and ϕε(x,m) := ϕan(x/ε,m) that oscillate periodically with period ε as the variable

x runs through Ω, describing the oscillations of the material dependent parameters of the composite. At

every scale ε, the energy associated to the ε-heterogeneous ferromagnet, will be given by the following

generalized Gibbs-Landau energy functional

GLε(m) :=

∫

Ω

aε|∇m|2 dτ +

∫

Ω

ϕε(·,m) dτ − µ0

2

∫

Ω

hd[Mεm] ·Mεm dτ −µ0

∫

Ω

ha ·Mεm dτ

=: Eε(m) =:Aε(m) =:Wε(m) =:Zε(m)

. (10.5)

The asymptotic Γ-convergence analysis of the family of functionals (GLε)ε∈R+ as ε tends to 0, is the object

of the present paper.

10.1.3 Statement of the main result

The main purpose of this paper is to analyze, by the means of both Γ-convergence and two-scale conver-

gence techniques, the asymptotic behavior, as ε→0, of the family of Gibbs-Landau free energy functionals

(GLε )ε∈R+ expressed by (10.5). Let us make the statement more precise. We consider the unit sphere S

2


136

of R3 and, for every s∈ S2, the tangent space of S

2 at a point s will be denoted by Ts(S2). The class of

admissible maps we are interested in is defined as

H1(Ω, S2) := m∈H1(Ω,R3) : m(x)∈ S2 for τ -a.e. x∈Ω,

where we have denoted by τ the Lebesgue measure on R3. We consider H1(Ω, S2) as a metric space

endowed with the metric structure induced by the classical L2(Ω, R3) metric. For every positive real

number t> 0, we set Qt := (0, t)3 and Q :=Q1 =(0, 1)3. We recall that a function u:R3→R is said to be

Q-periodic if u(·) =u(·+ei) for every ei in the canonical basis (e1, e2, e3) of R3.

For the energy densities appearing in the family (GLε)ε∈R+ we assume the following hypotheses:

H1. The exchange parameter aex is supposed to be a Q-periodic measurable function belonging

to L∞(Q) which is bounded from below and above by two positive constants cex > 0, Cex > 0,

i.e. 0< cex 6 aex(y) 6Cex for τ -a.e. y ∈ Q. In the setting of classical Calculus of Variations, this

hypothesis guarantees that the exchange energy density, which has the form g(y, ξ) := aex(y)|ξ |2,ξ ∈R

3×3, is a Carathéodory integrand satisfying the following quadratic growth condition for τ -

a.e. y ∈Q

∀ξ ∈R3×3 cex|ξ |2 6 g(y, ξ)6Cex(1+ |ξ |2).

Then we set aε(x)= aex(x/ε).

H2. The anisotropy density energy ϕan:R3× S2→R

+ is supposed to be a Q-periodic measurable

function belonging to L∞(Q) with respect to the first variable, and globally lipschitz with respect

to the second one (uniformly with respect to the first variable), i.e. ∃κL> 0 such that

ess supy∈Q

|ϕan(y,m1)− ϕan(y,m2)|6 κL|m1−m2| ∀m1,m2∈ S2. (10.6)

We then set ϕε(x,m) := ϕan(x/ε,m). The hypotheses assumed on ϕan are sufficiently general to

treat the most common cases of crystal anisotropy energy densities arising in applications. As a

sake of example, for uni-axial anisotropy, the energy density reads

ϕan(y,m) =κ(y)[1− (u(y) ·m)2], (10.7)

the spatially dependent unit vector u(·) being the easy axis of the crystal. For cubic type

anisotropy, the energy density reads as:

ϕan(y,m)= κ(y)∑

i=1

3

[(ui(y) ·m)2− (ui(y) ·m)4] (10.8)

the spatially dependent unit vectors ui(·) being the easy three mutually orthogonal axes of the

cubic crystal.

10.1 Introduction

137

H3. The saturation magnetization Ms is supposed to be a Q-periodic measurable function belonging

to L∞(Q), and we set Mε(·)=Ms(·/ε).

The main result of this paper is the following:

Theorem 10.1. Let (GLε)ε∈R+ be a family of Gibbs-Landau free energy functionals satisfying (H1),

(H2) and (H3). The family (GLε)ε∈R+ is equicoercive in the metric space (H1(Ω,S2), dL2(Ω,S2)). Moreover

(GLε)ε∈R+ Γ-converges in (H1(Ω, S2), dL2(Ω,S2)) to the functional Ghom:H1(Ω, S2)→R

+ defined by

Ghom(m) := Ehom(m)+Ahom(m) +µ0

2Whom(m)+ µ0Zhom(m). (10.9)

The four terms that appear in (10.9) have the following expressions:

Ehom(m) :=

∫

Ω

[

∫

Q

aex(y)|(I +∇ψ(y))ξ |2dy]

ξ=∇m(x)

,

the component of ψ being given, for j=1,2,3, as the unique solutions of the following scalar variational

problems

ψ · ej := argminϕ∈W#

1,∞(Q,R)

∫

Q

aex(y) |ej+∇ϕ(y)|2 dy.

The homogenized anisotropy energy is given by

Ahom(m) :=

∫

Ω

∫

Q

ϕan(y,m(x)) dydx,

while the homogenized magnetostatic self-energy is given by

Whom(m) :=−〈Ms〉Q2∫

Ω

hd[m] ·m dτ +

∫

Ω

∫

Q

|∇yϕm(x, y)|2 dxdy,

where, for every x ∈ Ω, the scalar function ϕm: Ω × Q→ R, is the unique solution of the following

variational cell problem:

m(x) ·∫

Q

Ms(y)∇yψ(y) dy=−∫

Q

∇yϕm(x, y) · ∇yψ(y) dy ,

∫

Q

ϕm(x, y) dy= 0

for every ψ ∈H#1 (Q).

Finally, the homogenized interaction energy is given by

Zhom(m) =−〈Ms〉Q∫

Ω

ha ·m dµ.

The paper is organized as follows. In Section 10.2 we give a survey of the concepts and results used

throughout the paper. The proof of Theorem 10.1 is established in Section 10.3 and more precisely: the

equi-coercivity of the family (GLε)ε∈R+ is established in Subsection 10.3.1; the Γ-limit of the exchange


138

energy family of functionals (Eε)ε∈R+ is computed in Subsection 10.3.2; in Subsection 10.3.3 it is shown

that the family of magnetostatic self-energies (Wε)ε∈R+ continuously converges to Whom, while in Sub-

section 10.3.4 it is established the continuous convergence of the family of anisotropy energies (Aε)ε∈R+

to Ahom; in Subsection 10.3.5, even if straightforward, is proved the continuous convergence of the family

of interaction energies (Zε)ε∈R+ to the functional Zhom. Finally, we conclude the proof of Theorem 10.1

in Subsection 10.3.6.

10.2 Preliminaries

The purpose of this section is to fix some notation and to give a survey of the concepts and results that

are used throughout this work. All results are stated without proof as they can be readily found in the

references given below.

10.2.1 Γ-convergence of a family of functionals

We start by recalling De Giorgi’s notion of Γ-convergence and some of its basic properties (see [DF75,

Dal93]). Throughout this part we indicate with (X, d) a metric space and, for every m∈X , with Cd(m)

the subset of all sequences of element of X which converge to m.

Definition 10.2. (Γ-convergence of a sequence of functionals) Let (Fn)n∈N be a sequence of

functionals defined on X with values on R. The functional F :X→R is said to be the Γ-lim of (Fn)n∈N

with respect to the metric d, if for every m∈X we have:

∀(mn)∈Cd(m) F(m) 6 liminfn→∞

Fn(mn) (10.10)

and

∃(mn)∈Cd(m) F(m)= limn→∞

Fn(mn). (10.11)

In this case we write

F = Γ- limn→∞

Fn. (10.12)

The condition (10.11) is sometimes referred in literature as the existence of a recovery sequence.

Definition 10.3. (Γ-convergence of a family of functionals) Let (Fε)ε∈R+ be a family of functionals

defined on X with values on R. The functional F :X→R is said to be the Γ-lim of (Fε)ε∈R+ with respect

to the metric d, as ε→ 0, if for every εn ↓ 0

F =Γ- limn→∞

Fεn. (10.13)

In this case we write F =Γ-limε→0Fε.Definition 10.4. A family of functionals (Fε)ε∈R+ is said to be equicoercive if there exists a compact

set K such that for each sequence εn ↓ 0,

infm∈X

Fεn(m) = inf

m∈KFεn

(m) ∀n∈N. (10.14)

One of the most important properties of Γ-convergence, and the reason why this kind of variational

convergence is so important in the asymptotic analysis of variational problems, is that under appropriate

compactness hypotheses it implies the convergence of (almost) minimizers of a family of equicoercive

functionals to the minimum of the limiting functional. More precisely, the following result holds.

10.2 Preliminaries

139

Theorem 10.5. (Fundamental Theorem of Γ-convergence) If (Fε)ε∈R+ is a family of equicoercive

functionals Γ-converging on X to the functional F . Then F is coercive and lower semicontinuous

(therefore there exists a minimizer for F on X) and we have the convergence of minima values

minm∈X

F(m)= limε→0

infm∈X

Fε(m). (10.15)

Moreover, given εn ↓ 0 and (mn)n∈N a converging sequence such that

limn→∞

Fεn(mn)= lim

n→∞

(

infm∈X

Fεn(m)

)

, (10.16)

its limit is a minimizer for F on X . If (10.16) holds, the sequence (mn)n∈N is said to be a sequence of

almost-minimizers for F .

Given two families of functional (Fε)ε∈R+ and (Gε)ε∈R+ Γ-converging respectively to F and G, is ingeneral not true that Γ-limε→0 (Fε+Gε)=F +G. A sufficient condition for this property to be true is that

at least one of the two families of functionals satisfy a stronger kind of convergence called continuous

convergence.

Definition 10.6. We say that a family of functionals (Gε)ε∈R+ is continuously convergent in X to a

functional G:X→R, and we will write Gε→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→ →Γcont G, if for every m0∈X and every real number η >0 there

exists an (ε0, δ0)∈R+×R

+ such that

∀ε∈R+ ∀m∈X (ε<ε0 and d(m,m0)<δ =⇒ | Gε(m)−G(m0) |< η).

We then have (see [Dal93] for a proof):

Proposition 10.7. Let F = Γ-limε→0Fε. Suppose that (Gε)ε∈R+ is continuously convergent to a

functional G, and that the functional Gε and G are everywhere finite on X . Then G = Γ-limε→0Gε and

Γ- limε→0

(Fε+ Gε)=F + G.

In particular if Z:X→R is a continuous functional then Γ-limε→0 (Fε+Z)=F +Z and Z is called a

continuous perturbation of the Γ-limit.

10.2.2 Two-scale convergence

The aim of this section is to present in a schematic way the main properties of two-scale convergence,

a notion independently introduced by Allaire [All92] and Nguetseng [Ngu89], and further developed by

many others (see [AB96] for instance).

We denote by C#∞(Q) the set of infinitely differentiable real functions over R

3 that are Q-periodic

and define H#1 (Q) as the closure of C#

∞(Ω) in H#1 (Ω). By C∞(Ω)⊗C#

∞(Q) we denote the set containing

all infinitely differentiable real functions over Ω×R3 that are Q-periodic in the second variable.

Definition 10.8.Given an open set Ω, a function φ∈L2(Ω× (R3/Q)) is said to be acceptable if

limε→0

∫

Ω

|φ(x, x/ε)|2 dx=

∫

Ω×Q

|φ(x, y)|2 dydx.


140

It is possible to prove (see [All92]) that a either L2(Ω, C#(Ω)) or L2(Q,C(Ω)) are acceptable spaces.

A generalized version of theRiemann-Lebesgue Lemma holds for the weak limit of rapidly oscillating

functions. For a proof we refer the reader to [CD99].

Lemma 10.9. Let Ω ⊂R3 be any open set. Let 1 6 p <∞ and t > 0 be a positive real number. Let

u∈Lp(Qt) be a Qt-periodic function. Set uε(x) := u(x/ε) µ-a.e. on Ω. Then, if p<∞, as ε→ 0

uε 〈u〉Qt:=

1

|Qt|

∫

Qt

u dτ weakly in Lp(Ω) .

If p=∞, one has

uε 〈u〉Qt:=

1

|Qt|

∫

Qt

u dτ weakly∗ in L∞(Ω) .

Definition 10.10. (Two-scale convergence) Given an open set Ω ⊂ R3, let E be a subspace of

L2(Ω× (R3/Z3)) such that every function in E is acceptable. A bounded family of function (uε)ε∈R+ in

L2(Ω) is said to E-two-scale converge to a limit u∈L2(Ω×Q), and we will write uε2su, if

limε→0

∫

Ω

uε(x)φ(x, x/ε) dx =

∫

Ω

∫

Q

u(x, y)φ(x, y) dydx (10.17)

for all φ∈E.

Finally we recall a simple criteria that justify the convergence of products (cfr. [All92]).

Proposition 10.11. If (uε)ε∈R+ and (vε)ε∈R+ are bounded families of functions belonging to L2(Ω)

that respectively two-scale converge to u and v in L2(Ω×Q), and if

‖u‖Ω×Q= liminfε→0

‖uε‖Ω

then

limε→0

∫

Ω

uε(x)vε(x)φ(x, x/ε) dx=

∫

Ω

∫

Q

u(x, y) v(x, y) φ(x, y) dxdy

for all φ∈C∞(Ω)⊗C#∞(Q).

10.3 The Homogenized Gibbs-Landau Free Energy Functional

This section is devoted to the proof of Theorem 10.1, i.e. to the proof of the equicoercivity of the family

of Gibbs-Landau free energy functionals (GLε )ε∈R+ expressed by (10.5), and to the identification of the Γ-

limits of the energy terms (Eε)ε∈R+, (Wε)ε∈R+, (Aε)ε∈R+ and (Zε)ε∈R+ arising in (GLε)ε∈R+. For the sake

of clarity, after a first section in which the equi-coercivity is analyzed, all different energy terms will be

separately studied in different sections.


141

10.3.1 The equicoercivity of the composite Gibbs-Landau free energy functionals

Equicoercivity has an important role in homogenization theory. In fact, the metric space in which to

work, must be able to guarantee the equicoercivity of the family of functionals under consideration, i.e.

the validity of the Fundamental Theorem of Γ-convergence.

Proposition 10.12. The family (GLε)ε∈R+ of Gibbs-Landau free energy functionals is equicoercive on

the metric space (H1(Ω, S2), dL2(Ω,S2)).

Proof. We recall that the general hypotheses on GLε impose the following bounds

∀y ∈Q 0<cex6 aex(y) 6Cex , 0 6Ms(y)6Cs , 06 ϕ(y,m)6Can.

Therefore, for every ε> 0

0 6 cex‖∇m‖Ω2 6 GL

ε(m) 6 Cex ‖∇m‖Ω2 +

µ0

2‖Mε‖Ω

2 +Can|Ω|+ ‖ha‖Ω‖Mε‖Ω

6 Cex‖∇m‖Ω2 +

µ0

2Cs

2|Ω|+Can|Ω|+Cs|Ω|1/2‖ha‖Ω

6 C⋆(1 + ‖∇m‖Ω2 )

where we set C⋆ = max

Cex,µ0

2Cs

2|Ω|, Cs|Ω|1/2‖ha‖Ω, Can|Ω|

. Since for every constant in space

magnetization u, GLε(u)6C⋆ for every ε> 0, we have

infm∈H1(Ω,S2)

GLε = inf

m∈KGLε ,

where K := m∈H1(Ω,S2) : GLε(m)6C⋆ is H1(Ω)-closed. The set K is not empty since every constant

in space magnetization m belongs to K. Moreover for every m∈K

‖m‖H1(Ω,S2)2 = ‖m‖Ω

2 + ‖∇m‖Ω2 6 |Ω|+ cex

−1GLε(m)6 |Ω|+ cex

−1C∗

so that K is (closed) and bounded in H1(Ω, S2). By Rellich’s compactness theorem the set K is a

compact subset of the metric space (H1(Ω, S2), dL2(Ω,S2)) and this implies that GL

ε is equicoercive in

(H1(Ω, S2), dL2(Ω,S2)).

10.3.2 The Γ-limit of exchange energy functionals Eε

The fundamental constraint of micromagnetic theory, i.e. the fact that the domain of definition of the

family Eε is a manifold value Sobolev space, requires more effort in the identification of the Γ-limit. In

what follows we make use of the following theorem due to Babadjian and Millot (see [BM10]) in which

the dependence of the Γ-limit from the tangent bundle of the manifold is taken into account via the so-

called tangentially homogenized energy density. We state the proposition in a bit less general form

which is adequate for our purposes.


142

Proposition 10.13. Let M be a connected smooth sub-manifold of R3 without boundary and g:

R3×R

3×3→R+ be a Carathéodory function such that

1. For every ξ ∈ R3×3 the function g(·, ξ) is Q-periodic, i.e. such that if (e1, e2, e3) denotes the

canonical basis of R3, one has

∀i∈ 1, 2, 3, ∀y ∈R3, ∀ξ ∈R

3×3 g(y+ ei, ξ) = g(y, ξ).

2. There exist 0<α6 β <∞ such that

α|ξ |2 6 g(y, ξ)6 β(1+ |ξ |2) for a.e. y ∈R3 and all ξ ∈R

3×3 .

Then the family

Eε(m) :=

∫

Ω

g(x/ε,∇m) dτ (10.18)

defined in the metric space (H1(Ω,M), dL2(Ω,M)) Γ-converges to the functional

Ehom(m) :=

∫

Ω

Tghom(m,∇m) dτ , (10.19)

where for every s∈M and ξ ∈ [Ts(M)]3,

Tghom(s, ξ)= limt→∞

(

infϕ∈W0

1,∞(Qt,Ts(M))

1

|Qt|

∫

Qt

g(y, ξ+∇ϕ(y)) dy

)

, Qt := (0, t)3 (10.20)

is the tangentially homogenized energy density.

We refer the reader to [BM10] for the more general version and the proof. Let us nevertheless empha-

size why the tangent bundle TM plays a role. In order to understand this, it is convenient to develop a

minimizer mε of Eε under the so-called multiscale expansion

mε(x)=m0(x)+εm1(

x,x

ε

)

+ o(ε) (10.21)

where m0,m1 are respectively a minimizer of the Γ-limit and the first order corrector. Clearly, due to

the constraintmε(x)∈M for a.e. x∈Ω, we get, passing formally to the limit,m0(x)∈M for a.e. x∈Ω,

while the corrector m1(x, y) ∈ Tm1(x)M for a.e. y ∈ Q. Plugging (10.21) into (10.18) formally leads to

(10.19) where the energy density is defined by (10.20).

Let us go back to the application of this result in our setting. We consider the family of exchange

energy functionals, all defined in H1(Ω, S2), given by (Eε)ε∈R+. Since (H1) holds, Proposition 10.13

ensures that the family (Eε)ε∈R+ Γ-converges in the metric space (H1(Ω; S2), dL2(Ω,S2)), i.e. with respect

to the topology induces on H1(Ω, S2) by the strong L2(Ω,R3) topology, to the functional

Ehom:H1(Ω, S2) → R+

m 7→ Ehom(m)=

∫

Ω

Tghom(m,∇m) dτ


143

where for every s∈S2 and every ξ ∈ [Ts(S

2)]3,

Tghom(s, ξ) = limt→∞

[

infϕ∈W0

1,∞(Qt,Ts(S2))It[ξ,ϕ]

]

(10.22)

with

It[ξ,ϕ] :=1

|Qt|

∫

Qt

aex(y)|ξ+∇ϕ(y)|2 dy. (10.23)

Equivalently (see [BD98]), into the periodic setting

Tghom(s, ξ) = limt→∞

[

infϕ∈W#

1,∞(Qt,Ts(S2))It[ξ,ϕ]

]

, (10.24)

and from the convexity of the integrand, it is routinely seen that we can replace the limit for t→∞ by the

computation on the unit cell (though still with periodic boundary conditions). We are therefore left with

Tghom(s, ξ) = infϕ∈W#

1,∞(Q,Ts(S2))

∫

Q

aex(y)|ξ+∇ϕ(y)|2 dy. (10.25)

It can be proved, by means of Lax-Milgram theorem, that for every s∈S2 and every ξ ∈ [Ts(S

2)]3 there

exists a unique solution ϕ(s, ξ) of this latter problem, up to an additive constant that we may fix by

requiring moreover that 〈ϕ〉Q=0. Moreover, for every s∈S2, the map ξ ∈ [Ts(S

2)]3 7→ϕ(s, ξ)∈W#1,∞(Q,

Ts(S2)) is a linear map.

Let us now consider the problem that defines the classical homogenization tensor, namely

ghom(ξ) = infϕ∈W#

1,∞(Q,R3)

∫

Q

aex(y)|ξ+∇ϕ(y)|2 dy. (10.26)

It is well known that ghom is actually a quadratic form in ξ and that there exists a symmetric and positive

definite matrix Ahom∈R3×3 such that

ghom(ξ)= (Ahomξ1, ξ1) + (Ahomξ2, ξ2) + (Ahomξ3, ξ3). (10.27)

Since in this latter problem, the space of functions among which minimization takes place is bigger than

the one for the original problem, one clearly has

ghom(ξ)6Tghom(s, ξ) .

To prove the equality of the infima it is therefore sufficient to show that for every ϕ∈W#1,∞(Q,R3) there

exists a couple (s,ψ)∈S2× [Ts(S2)]3 such that∫

Q

aex(y)|ξ+∇ψ(y)|2 dy6

∫

Q

aex(y)|ξ+∇ϕ(y)|2 dy. (10.28)

To this end we observe that if if ϕ= (ϕ1, ϕ2, ϕ3) is a solution to (10.26), denoting by ψ := ϕ− (ϕ · s)sthe nearest point projection of ϕ on [Ts(S

2)]3, one has, since ξ ∈ [Ts(S2)]3,

∫

Q

aex(y)|ξ+∇ψ(y)|2 dy=

∫

Q

aex(y)|ξ+∇ϕ(y)|2 dy−∫

Q

aex(y)|∇(ϕ · s)(y)|2 dy ,

and therefore, since the second term in the previous equation is less or equal then zero, relation (10.28)

holds.


144

Moreover, if ϕ = (ϕ1, ϕ2, ϕ3) is a solution to (10.26), denoting by ψ := ϕ − (ϕ · s)s, one has, since

ξ ∈ [Ts(S2)]3

∫

Q

aex(y)|ξ+∇ϕ(y)|2 dy=

∫

Q

aex(y)|ξ+∇ψ(y)|2 dy+

∫

Q

aex(y)|∇(ϕ · s)(y)|2 dy .

Notice that s is a constant which in particular does not depend on y in this latter formula. This in

particular shows that for every s∈S2 and for every ξ∈ [Ts(S

2)]3 any solution ϕ of (10.26) actually satisfies

ϕ · s= 0, and is therefore a solution to (10.25). We therefore deduce that

ghom(ξ)=Tghom(s, ξ) .

and in particular, that Tghom(s, ξ) does not depend on s, and is given by (10.27).

10.3.3 The continuous limit of magnetostatic self-energy functionals Wε

In what follows we will make use of the following proposition (see [San07] for a proof).

Proposition 10.14. Let (mε)ε∈R+ be a bounded sequence in L2(R3,R3) that two-scale converges to

m(x, y). Then the two-scale limit of (hd[mε])ε∈R+ is given by

hd(x, y)=hd

[

∫

Q

m(x, y) dy

]

+∇yϕ(x, y)

where for every x ∈ R3 the scalar function ϕ(x, ·) is the unique solution in H#

1 (Q) of the variational

problem

∫

Q

m(x, y) · ∇yψ(y) dy=−∫

Q

∇yϕ(x, y) · ∇yψ(y) dy ,

∫

Q

ϕ(x, y) dy=0 (10.29)

for all ψ ∈H#1 (Q).

In particular in the context of our problem we get the following

Corollary 10.15. Let (mε)ε∈R+ =(Mεm)ε∈R+ then the family of magnetostatic self-energies

Wε:m∈L2(Ω, S2) 7→−(hd[Mεm],Mεm)Ω

continuously converges to the functional

Whom:m∈L2(Ω, S2) 7→−〈Ms〉Q2 (hd[m],m)Ω + ‖∇yϕm‖Ω×Q2

where for every x ∈ Ω the scalar function ϕm: Ω × Q → R is the unique solution of the following

variational cell problem:

m(x) ·∫

Q

Ms(y)∇yψ(y) dy=−∫

Q

∇yϕm(x, y) · ∇yψ(y) dy ,

∫

Q

ϕm(x, y) dy= 0

for all ψ ∈H#1 (Q).


145

Proof. Since Mε(x)m(x) ε→0

2sMs(y)m(x) for every m ∈L2(Ω, S2), from Proposition 10.11 and Propo-

sition 10.14 we get

limε→0

−(hd[Mεm],Mεm)Ω =−〈Ms〉Q2 (hd[m],m)Ω−∫

Ω×Q

∇yϕm(x, y) ·Ms(y)m(x) dxdy.

Indeed for every x ∈ Ω the scalar function ϕ(x, ·) is the unique solution of the variational cell problem

(10.29), therefore setting ψ(·) := ϕ(x, ·) in (10.29) we get

−m(x) ·∫

Q

Ms(y)∇yϕ(x, y) dy=

∫

Q

|∇yϕm(x, y)|2 dy

and therefore

limε→0

−(hd[Mεm],Mεm)Ω =−〈Ms〉Q2 (hd[m],m)Ω + ‖∇yϕm‖Ω×Q2 =Whom(m) (10.30)

Now we show that the family Wε→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→ →Γcont

ε→0Whom. We have to prove that for every m0∈L2(Ω,S2) and every

η > 0 there exists (ε0, δ)∈ (R+×R+) such that

∀m∈L2(Ω, S2) ∀ε∈R+ (ε<ε0 and ‖m−m0‖Ω<δ =⇒ |Wε(m)−Whom(m0) |< η).

To this end we start by observing that the correspondence m 7→ ∇yϕm is a linear and continuous map

from L2(Ω,R3) to L2(Ω×Q,R3) with

‖∇yϕm‖Ω×Q 6cM ‖m‖Ω (with cM := ‖Ms‖Q).

Therefore, for every m,m0∈L2(Ω, S2)

|Wε(m)−Whom(m0)| 6 |Wε(m)−Whom(m)|+ |Whom(m)−Whom(m0)|6 |Wε(m)−Whom(m)|+ 〈Ms〉Q2 |(hd[m],m)Ω− (hd[m0],m0)Ω|

+ |‖∇yϕm‖Ω×Q2 −

‖∇yϕm0‖Ω×Q2 |

= |Wε(m)−Whom(m)|+ 〈Ms〉Q2 |(hd[m+m0],m−m0)Ω|+ |(∇yϕm+m0

,∇yϕm−m0)Ω×Q|

6 |Wε(m)−Whom(m)|+ (〈Ms〉Q2 + cM2 )‖m+m0‖Ω‖m−m0‖Ω

6 |Wε(m)−Whom(m)|+ 2|Ω|1/2(〈Ms〉Q2 + cM2 )‖m−m0‖Ω

By (10.30) follows the existence of a sufficiently small ε0 such that

∀ε<ε0∣

∣Wε(m)−Whom(

m)∣

∣<η

2.

On the other hand for every m∈L2(Ω, S2) such that ‖m−m0‖Ω< η(

4|Ω|1/2(〈Ms〉Q2 + cM2 ))

−1 we get

|Wε(m)−Whom(m0) | 6 η,


146


10.3.4 The continuous limit of the anisotropy energy functionals Aε

Proposition 10.16. If the anisotropy energy density ϕ: R3 × S2 →R

+ is Q-periodic with respect to

the first variable and globally lipschitz with respect to second one (uniformly with respect to the first

variable) then

(

Aε:m∈L2(Ω, S2) 7→∫

Ω

ϕ(x/ε,m(x)) dx

)

→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→ →Γcont

ε→0

(

Ahom:m∈L2(Ω, S2) 7→∫

Ω

∫

Q

ϕ(y,m(x)) dydx

)

.

Proof. We have to prove that for every m0∈L2(Ω, S2) and every η > 0 there exists (ε0, δ)∈ (R+×R+)

such that

∀m∈L2(Ω, S2) ∀ε∈R+ (ε< ε0 and ‖m−m0‖Ω<δ =⇒ |Aε(m)−Ahom(m0) |< η).

We observe that for every m,m0∈L2(Ω, S2) we have

∣

∣

∣

∣

∫

Ω

ϕ(x/ε,m(x))−〈ϕ(·,m0(x))〉Q dx

∣

∣

∣

∣

6

∣

∣

∣

∣

∫

Ω

ϕ(x/ε,m(x))−〈ϕ(·,m(x))〉Q dx

∣

∣

∣

∣

+

∫

Ω

| 〈ϕ(·,m(x))〉Q−〈ϕ(·,m0(x))〉Q | dx(10.31)

Since (cfr. Lemma 10.9)

ϕ(x/ε,m(x)) 〈ϕ(·,m(x))〉Q=

∫

Q

ϕ(y,m(x)) dy weakly∗ in L∞(Ω),

there exists a sufficiently small ε0 such that

∀ε< ε0∣

∣

∣

∣

∣

∫

Ω

(

ϕ(x/ε,m(x))−∫

Q

ϕ(y,m(x)) dy

)

dx

∣

∣

∣

∣

∣

<η

2. (10.32)

On the other hand, by the lipschitz condition hypothesis

∫

Ω

∫

Q

|ϕ(y,m(x))−ϕ(y,m0(x))| dydx 6 cL

∫

Ω

|m(x)−m0(x)| dx 6 cL|Ω|1/2‖m−m0‖Ω . (10.33)

Substituting estimates (10.32) and (10.33) into (10.31) we get

| Aε(m)−Ahom(m0) | 6 η

2+ c⋆‖m−m0‖Ω

(

with c⋆ := cL|Ω|1/2)

Therefore for every m∈L2(Ω, S2) such that ‖m−m0‖Ω< η/(2c⋆) we get

| Aε(m)−Ahom(m0) |6η,


147


Corollary 10.17. (Homogenized uni-axial anisotropy energy density). If ϕ(y,m) = κ(y)|m∧u(y)|2 then

Ahom(m)=

∫

Ω

〈κ〉Q−〈κu⊗u〉Q :m⊗m dτ .

10.3.5 The continuous limit of the interaction energy functionals Zε

The convergence of (Zε)ε∈R+ to Zε is straightforward. Indeed this energy term is expressed by the

product, with respect to the L2(Ω) scalar product, of the constant function ha and the weakly converging

sequence (Mεm)ε∈R+ 〈Ms〉Q weakly∗ in L∞(Ω) (cfr. Lemma 10.9). Therefore repeating the same

argument given in the previous Subsection:

Zε→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→ →Γcont

ε→0Zhom with Zhom(m) :=−〈Ms〉Q

∫

Ω

ha ·m dτ .

10.3.6 Proof of Theorem 10.1 completed

It is now easy to finish the proof of Theorem 10.1. Indeed it is sufficient to recall (Proposition 10.7) the

stability of Γ-limit under the sum of continuously convergent families of functionals. In fact, what has

been proved in the previous subsections, can be summarized in the following convergence scheme

Eε →→→→→→→→→→Γ

ε→0Ehom , Wε→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→ →Γcont

ε→0Whom , Aε→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→ →Γcont

ε→0Ahom , Zε→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→→ →Γcont

ε→0Zhom,

and Proposition 10.7 concludes the proof.


148

11Basic equations for Magnetization Dynamics

11.1 The Landau-Lifshitz-Gilbert equation for magnetization dynamics.

As we have seen in Chapter 7, when m×heff[m]=/ 0, i.e. when the Brown’s static equation (7.25) is not

satisfied, the system is not at equilibrium and will evolve in time according to some appropriate dynamic

equation.

11.1.1 The Landau-Lifshitz equation.

The equation originally proposed by Landau and Lifshitz [LL35] is mostly used for the description of

magnetization dynamics. This equation is based on the idea that in a ferromagnetic body the effective

field heff[m] will induce a precession of local magnetization m of the form:

∂m

∂t=−m×heff[m]. (11.1)

The dynamics described by equation (11.1) is such that the magnitude (length) of magnetization is

conserved. Indeed, every solution m: (x, t)∈Ω× I→R3 of (11.1) is such that

∀x∈Ω1

2

∂ |m(x, t)|2∂t

=m(x, t) · ∂m∂t

(x, t)=−m · (m×heff[m])= 0, (11.2)

therefore if |m(x, t0)|=ms(x) for some t0∈ I and every x∈Ω, then |m(x, t)|=ms(x) ∀x∈Ω, ∀t∈ I. Inparticular, if |m(x, t0)|=1 then |m(x, t)|=1 for every t∈ I. Thus, equation (11.1) is consistent with the

fundamental micromagnetic constraint (6.43).

However equation (11.1) cannot describe any approach to equilibrium resulting in energy decrease

due to interaction with a thermal bath. Indeed under suitable regularity assumptions, by the chain rule,

we can compute the rate of energy change by starting from the general relation:get the general relation

∂tGL(m) =− 1

|Ω|

∫

Ω

heff[m] · ∂m∂t

dτ (11.3)

149

which is valid assuming that the homogeneous Neumann boundary condition ∂nm=0 is valid during the

dynamics. Taking into account equation (11.1), we get

∂tGL(m)= 0. (11.4)

which means that the dynamics is non-dissipative.

Energy relaxation mechanisms can be taken into account by introducing an additional phenomenolog-

ical term chosen through heuristic considerations. In their original paper, Landau and Lifshitz described

damping by a term proportional to the component of heff that is perpendicular to the magnetization:

∂m

∂t=−m×heff[m] +α[heff[m]− (m ·heff[m])m]. (11.5)

Here, α is a dimensionless non negative constant called damping constant. Its value is quite small, of

the order of 10−4, 10−3 in garnets and of the order of 10−2 in cobalt or permalloy.

The rationale behind equation (11.5) can be explained as follows. The effective field heff identifies, in

the space of possible configurations for m, the direction of steepest energy decrease, so it would be the

natural direction for magnetization relaxation. However, the magnetization magnitude must be preserved

as well. This suggests that only the heff component perpendicular to m may contribute to ∂tm. It is

apparent that this component coincides with the vector −m × (m × heff[m]). Consequently, equation

(11.5) can be written in the equivalent form:

∂m

∂t=−m×heff[m] −αm× (m×heff[m]). (11.6)

This is the form in which the Landau-Lifshitz equation is mostly used in the literature.

heff[m]

m

heff[m]

m

Non-dissipative DissipativeLL equation LL equation

Figure 11.1. (left) Landau-Lifshitz equation cannot describe any approach to equilibrium resulting in energydecrease due to interaction with a thermal bath. (right) Energy relaxation mechanisms can be taken into accountby introducing an additional phenomenological term chosen through heuristic considerations.

Basic equations for Magnetization Dynamics

150

Remark 11.1. There is no a priori reason for assuming that the damping constant α should be constant.

In general it can be a function of the state of the system. It is only for the sake of simplicity that this

quantity is assumed to be a constant parameters in most studies of magnetization dynamics.

One can compute the rate of energy change by starting from the general relation (11.3): indeed,

substituting (11.5) into (11.3) we find that

∂tGL(m)=− α

|Ω|

∫

Ω

|m×heff[m]|2 dτ (11.7)

which shows that the energy is always a decreasing function of time. This property is often referred to

as Lyapunov structure of the Landau-Lifshitz equation with damping.

11.1.2 The Landau-Lifshitz-Gilbert equation.

Another equation for the description of magnetization dynamics in ferromagnets has been proposed by

Gilbert [Gil04]. This equation has the form:

∂m

∂t=−m×heff[m] +αm× ∂m

∂t. (11.8)

and is consistent with the fundamental micromagnetic constraint (6.43): it is sufficient to dot multiply

both members by m to recognize this.

Equation (11.8) deserves special attention, because it can be derived from a suitable Lagrangian

formulation of magnetization dynamics and a Rayleigh dissipation function. By writing equation (11.8)

in the form:

∂m

∂t=−m×

(

heff[m]−α∂m

∂t

)

(11.9)

we observe that, in the Gilbert equation, relaxation to equilibrium is accounted for by subtracting from the

effective field a viscous-type term proportional to the time derivative of magnetization. This is reflected

also in the equation for the energy balance. Substituting equation (11.8) into the general relation (11.3)

we get

∂tGL(m) =− α

|Ω|

∫

Ω

heff[m] ·(

m× ∂m

∂t

)

dτ =α

|Ω|

∫

Ω

∂m

∂t· (m×heff[m]) dτ . (11.10)

On the other hand, from (11.9) we get

∂m

∂t· (m×heff[m]) =−

∣

∣

∣

∣

∂m

∂t

∣

∣

∣

∣

2

(11.11)

and therefore, instead of equation (11.7), we now obtain the relation:

∂tGL(m)=− α

|Ω|

∫

Ω

∣

∣

∣

∣

∂m

∂t

∣

∣

∣

∣

2

dτ (11.12)

11.1 The Landau-Lifshitz-Gilbert equation for magnetization dynamics.

151

The previous property is often referred to as Lyapunov structure of the Landau-Lifshitz-Gilbert equa-

tion.

The Landau-Lifshitz-Gilbert equation (11.8) is mathematically equivalent to the Landau-Lifshitz

equation (11.6). The equivalence is readily proven by vector multiplying both sides of (11.6) by m, and

by using the identity: m× (m× (m×heff[m]))=−m×heff[m]. This leads to the formula:

m× (m×heff[m])=−m× ∂m

∂t+ αm×heff[m] (11.13)

which substituted back in (11.6) gives

∂m

∂t=−(1 +α2)m×heff[m] +αm× ∂m

∂t. (11.14)

If we now suppose that the two time scales appearing in equations (11.8) and (11.6), are not the same,

but proportional: i.e. that for some constant γG

(

∂m

∂t

)

L

= γG

(

∂m

∂t

)

G

, (11.15)

then the Landau-Lifshitz equation (11.6) coincides with the Gilbert equation (11.8) provided that

γG=(1+α2). (11.16)

11.2 Spatially uniform magnetization dynamics.

In many technological applications, where the size of the magnetic media has reached the nanometer

scale, it is reasonable to assume that the exchange interaction is prevalent with respect to the others

and, therefore, that the particle tends to be uniformly magnetized. In other words, the uniform mode

is energy-favored with respect to disuniformities as soon as the characteristic dimension of the body is

comparable or even smaller than the exchange length. In this framework, it does make sense to neglect

non-uniform modes and consider the particle as uniformly magnetized.

The uniform dynamics is governed by the Landau-Lifshitz-Gilbert equation (11.8) which for the case

of spatial magnetization uniformity reads as

dm

dt=−m×heff[m] +αm× dm

dt. (11.17)

Since exchange energy gives zero contribution to the free energy, the effective field heff[m] and the free

energy functional GL have now the simple expressions (see (7.19) and (6.44))

heff[m] :=hd[m] +han(m)+h0 (11.18)

and

GL(m,h0)=1

|Ω|

∫

Ω

ϕan(m)− 1

2hd[m] ·m−h0 ·m dτ . (11.19)


152

In what follows we will assume that the region Ω occupied by the ferromagnetic particle is a general

ellipsoid with semi-axes aligned along the standard unit vectors e1, e2, e3 and semi-axes lengths a1 >

a2 > a3. Moreover we will assume uni-axial anisotropy along the major semi-axis direction e3. In these

hypotheses the magnetostatic field can be expressed in terms of the demagnetizing tensor (see Chapter

5) as

hd[m] =−

N1 0 00 N1 00 0 N3

m1

m2

m3

=:−Nd[m] (11.20)

where the demagnetizing factors N1,N2,N3 are such that N1+N2 +N3 =1. Therefore the magnetostatic

self-energy assume the form

W(m)=1

2Nd[m] ·m . (11.21)

The assumption of uni-axial anisotropy implies that the corresponding energy term is quadratic. Indeed,

if the easy axis is aligned along e3, then

A(m) :=1

|Ω|

∫

Ω

ϕan(m)dτ = κ1|m× e1|2 = κ1(m22 +m3

2) (11.22)

and therefore

han(m)=−∇ϕan(m)= 2κ1(m · e2)e2 + 2κ1(m · e3)e3 . (11.23)

Under these assumptions the expressions for the effective field heff[m] and the free energy functional GL

become

heff[m] :=−Nd[m]+ 2κ1(m2e2 +m3e2)+h0 =−

D1 0 00 D2 00 0 D3

m1

m2

m3

+h0 =−Dd[m] +h0 (11.24)

and

GL(m,h0) =1

22κ1m1

2 +1

22κ1m2

2 +1

2Nd[m] ·m−h0 ·m =

1

2Dd[m] ·m−h0 ·m (11.25)

where the coefficients D1 :=N1,D2 :=N2−2κ1 and D3 :=N3−2κ1 take into account shape and crystalline

anisotropy.

The study of uniform magnetization dynamics has considerable simplifications as far as the mathe-

matical model is concerned. Indeed, in this framework, the LLG equation defines a dynamical system

evolving on the unit-sphere S2 due to fundamental micromagnetic constraint (6.43). In this respect the

LLG equation (11.17) describes an autonomous dynamical system whose phase space is 2D, and therefore,

it cannot exhibit chaotic behavior [Per00]. Moreover, by recalling the Lyapunov structure (11.12) of LLG

equation for constant field, which states that energy is a decreasing function of time (α > 0) one can

immediately conclude that the only steady solutions are fixed points. The number of these fixed points

is at least two and in any case is even, due to Poincaré index theorem [Per00]. Thus, any bifurcation of

fixed points involves two equilibria at the same time. The fixed points of the dynamics can be computed

from the Brown’s static equation

m×heff[m] =λm, |m|= 1 (11.26)

in the four scalar unknowns m = (m1, m2, m3) and λ. We refer the reader to [d’A04] and [BMS09] for

further details.

11.2 Spatially uniform magnetization dynamics.

153

11.2.1 Magnetization switching process

By the means of dynamical systems theory, some relevant technological applications connected with

magnetic recording devices can be addressed. In particular, in this subsection, we will focus our attention

on magnetization reversal processes, commonly referred to as magnetization switching processes.

Basically, if we assume that initially the magnetization vector m0 is aligned with some easy axis of

magnetization, then switching process consists in manipulating the control variables in order to let the

magnetization vector undergo a complete change of direction, i.e. m0 7→ −m0. Suppose we associate

the initial magnetization vector, at a certain time t0 when the magnetization is parallel to some easy

axis of magnetization, as a bit «0» then, when the switching process takes place one can then map that

phenomenon as a transition from the bit «0» to the bit «1». If we can control this phenomenon then we

found a way of processing digital information. The locality of this phenomenon is a scientific improvement

since it allows, in terms of digital information storage devices, to have more precision and thus one can

have larger areal densities and store more information.

The conventional way obtains the switching by using magnetic field produced by external currents, and

this technique is mostly used in hard disks realizations. Recently, the possibility of using spin-polarized

currents, injected directly into the ferromagnetic medium, has been investigated both experimentally and

theoretically. This way to control switching has considerable applications in MRAMs technology, since

in this way it is possible to circumvent the difficulty of generating magnetic fields that switch only the

target cell. The spin-polarized current driven switching will be analyzed in the next section.

11.3 Spin-momentum transfer in magnetic multilayers: Landau-Lifshitz-Slonczewski equation.

It has been recently shown, both theoretically and experimentally, that a spin-polarized current when

passing through a small magnetic conductor can affect its magnetization state. The interaction between

spin polarized current and magnetization in small ferromagnetic bodies can produce the switching of

magnetization direction [Slo96, Sun99]. This kind of dynamical behavior have potential applications in

magnetic storage technology and spintronics. In this respect, it was predicted, and later confirmed, that

spin-polarized current can lead to current-controlled switching in magnetic nanostructures.

I(t)

m

ha

z

xFerromagnetic

Free Layer

Nonmagnetic

Conductor

Ferromagnetic

Fixed Layer

p

Figure 11.2. Sketch of a uni-axial trilayer spin-valve.


154

Magnetization dynamics is described by the Landau-Lifshitz-Gilbert equation (11.17) and the effect

of spin-polarized currents can be taken into account through the additional torque term derived by

Slonczewski in his seminal paper [Slo96]. This model can be applied to describe the magnetization

dynamics in the free layer of trilayers structures constituted by two ferromagnetic layers separated by

nonmagnetic metal layer (typically the system is a Co-Cu-Co trilayers as sketched in Figure 11.2). One of

the ferromagnetic layer is «fixed», namely has a given and constant value of magnetization (denoted by p

in Figure 11.2) while the second ferromagnetic layer is a thin film where the magnetization is «free» to

change and where dynamics takes place. This kind of structure is traversed by an electric current whose

direction is normal to the plane of the layer (generally this configuration called current perpendicular

to plane (CPP) geometry). The fixed layer is instrumental to provide a controlled polarization (on

the average parallel to the fixed magnetization direction) of the electron spins which travel across the

trilayers, from the fixed to the free layer. It important to underline that the effect of spin induced torque

is predominant on the effect of the magnetic field generated by the current itself for structures which

have small enough transversal dimensions. By using reasonably estimate it has been predicted and then

verified experimentally that the effect of the current generated magnetic field can be considered negligible

for transversal dimension as small as 100nm.

11.3.1 Landau-Lifshitz-Gilbert equation with Slonczewski spin-transfer torque term.

In order to introduce a model equation for magnetization dynamics in presence of spin polarized currents,

let us first consider the model derived by Slonczewski in [Slo96]. In his paper, a five layers structure is

considered. In this structure, the first, the third and the fifth layers are constituted by paramagnetic

conductors and the second and the fourth layers are ferromagnetic conductors (it is a three layers structure

as the one mentioned in the introduction with paramagnetic conductors as spacer and contacts).

The multilayers system is traversed by an electric current normal to the layers plane. The electron

spins, polarized by the fixed ferromagnetic layer (the second layer) are injected by passing through the

paramagnetic spacer into the free ferromagnetic layer (the forth layer) where the interaction between

spin polarized current and magnetization takes place. The magnetic state of the ferromagnetic layers is

described by two vectors S1 and S2 representing macroscopic (total) spin orientation per unit area of the

fixed and the free ferromagnetic layers, respectively.

The connection of this two vectors with the total spin momenta L1 and L2 (which have the dimension

of angular momenta) is given by the equations L1=~S1A, L2=~S2A, where A is the cross-sectional area

of the multilayers structure. By using a semi-classical approach to treat spin transfer between the two

ferromagnetic layers, Slonczewski derived the following generalized LLG equation [Slo96]:

dS2

dt= s2×

(

γHuc ·S2c−αdS2

dt+ e−1Iegs1× s2

)

(11.27)

where s1, s2 are the unit vectors along S1, S2, γ is the absolute value of the gyromagnetic ratio, Hu is

the anisotropy field constant, c is the unit vector along the anisotropy axis (in-plane anisotropy), α is

the Gilbert damping constant, Ie the current density (electric current per unit surface), e is the absolute

value of the electron charge, g a scalar function given by the following expression

g(s1 · s2)=

[

(1+P )3(3 + s1 · s2)

4P 3/2− 4

]

−1

(11.28)

11.3 Spin-momentum transfer in magnetic multilayers: Landau-Lifshitz-Slonczewski equation.

155

and P is the spin polarizing factor of the incident current which gives the percent amount of electrons

that are polarized in the p direction (see [Slo96] for details).

The current Ie in (11.27) is assumed to be positive when the charges move from the fixed to the free

layer. Let us notice that in equation (11.27) the ferromagnetic body is assumed to be uni-axial with

anisotropy axis along c.

In the sequel, we will remove this simplifying assumption by taking into account the effect of the

strong demagnetizing field normal to the plane of the layer in order to consider the thin-film geometrical

nature of the free layer. Our next purpose is to derive from equation (11.27) an equation for magnetization

dynamics. We will carry out this derivation by using slightly different notation and translating all the

quantities in practical MKSA units.

By measuring the time t in units of (γMs)−1, and introducing the following definitions,

heff[m] :=κan(ex ·m)ex+hd[m] +ha, β(m) :=JeJpb(m) (11.29)

equation (2.91) can be written in the compact form

dm

dt−αm× dm

dt=−m× (heff[m] + βm× p). (11.30)

The term heff[m] + βm × p is known in literature as the generalized effective field and sometimes

denoted by Heff[m]. Equation (11.30) is is formally identical to LLG when there are no current-driven

torque term. The micromagnetic equilibria including spin torque effect are now related to the following

equations similar to Brown’s static equations (11.26)

m× (heff[m] + βm× p)=λm, |m|= 1. (11.31)

The basic difference between the ordinary effective field heff[m] and the generalized effective field Heff[m]

is that the first one can be derived by the gradient of a free energy, while the second one cannot.


156

12Current-driven microwave-assisted

Magnetization Switching

This chapter is devoted to the presentations of the results exposed in [dDS+11] where the switching

process of a uniformly magnetized spin-valve is investigated. The system is subject to external DC applied

fields and injected radio-frequency (RF) spin-polarized currents. The possibility of using the RF power

to obtain a reduced coercivity of the particle is related to the onset of chaotic magnetization dynamics

for moderately low values of the RF current amplitude. Perturbation technique for the estimation of

the reduced coercive field is developed and applied to the microwave assisted switching of the particle.

Numerical simulations confirm the predictions of the theory.

12.1 Introduction

One of the fundamental issues connected with the downscaling of magnetic storage devices is the thermal

stability of magnetization states. This problem can be circumvented by increasing the materials magnetic

anisotropy, but as a consequence, high applied magnetic fields are required to reverse the magnetization

states. For this reason, considerable attention has been recently paid to design new strategies of magne-

tization switching in which the applied field is assisted by additional external actions. Examples of these

new approaches are heat and microwave assisted switching [KGK+00, TWM+03, BMS+09, WB07].

In this chapter, by combining numerical and analytical techniques, we investigate the role of microwave

spin-polarized electric current in assisting magnetization switching of an anisotropic magnetic particle

(see Figure 11.2). The presence of a microwave spin-polarized current produces a reduction of the coercive

field associated with the particle anisotropy. In this respect, we analyze how this reduction is connected to

chaotic magnetization dynamics which may be induced by the microwave injected current. The presence

of chaotic dynamics is predicted by an an analytical technique based on the Melnikov-Poincaré integral.

157

ha

m

ep

I(t)

x

z

Figure 12.1. Sketch of a uni-axial trilayer spin-valve.

12.2 The analytical results

We consider a magnetic spin-valve-like nanostructure, subject to both DC fields and microwave injected

currents. Magnetization dynamics is governed by the Landau-Lifshitz- Slonczewski (LLS) equation [Slo96,

BMS09] which in normalized form reads:

dm

dt=−m×heff[m]−αm× (m×heff[m])+ β(t)m× (m× ep) (12.1)

where m(t) and ep are unit vectors associated with the orientations of magnetization in the free and in

the fixed layer of the spin-valve, respectively. In equation (12.1) time is measured in units of (γMs)−1

(γ is the absolute value of the gyromagnetic ratio), heff[m] is the effective field and α is the damping

constant, which is typically a positive quantity small with respect to unity: α≪ 1.

The function β(t) is proportional to the injected current I(t) through the relationship

β(t) =bpI(t)

SJp(12.2)

where bp is a model-dependent parameter in the order of unity, S is the device cross sectional area, and

Jp := µ0Ms2|e|d/~ is a characteristic current density (µ0 is the vacuum permeability, e is the electron

charge, d is the thickness of the free layer, and ~ is the reduced Planck constant). We assume that the

injected current is time-harmonic β(t):=βaccos (ωt), where βac and ω are the amplitude and the frequency

of the excitation, respectively. It is important to point out that, in typical experimental conditions, the

normalization current SJp/bp is considerably larger than the injected current I(t) and thus βac≪ 1.

The effective field heff[m] takes into account shape, crystalline anisotropy and Zeeman interaction,

and is given by the negative gradient heff[m] :=−∂mGL(m) of the particle free energy

GL(m)=1

2Dxmx

2 +1

2Dymy

2 +1

2Dzmz

2 − haxmx (12.3)

Current-driven microwave-assistedMagnetization Switching

158

where Dx<Dy<Dz are effective anisotropy constants and hax is the DC (constant) external field applied

in the easy axis direction.

In the absence of the injected current, the switching of magnetization from the state mx = −1 to

mx= +1 occurs at the coercive field hc of the particle given by hc=Dy −Dx. The situation is changed

when a microwave current is injected; in fact, the switching field can be considerably less than hc. We

want to analyze in details how this reduction is connected to the presence of chaotic dynamics [dSB+09].

It turns out that, for a given value of hax, the magnetization dynamics may exhibit chaotic behavior

and this circumstance is amenable of an analytical treatment. This is based on the fact that in equation

(12.1) the non-conservative terms, controlled by α and βac, are small quantities, which are assumed to

be of the same order of magnitude. Thus, equation (12.1) can be written in the following perturbative form

dm

dt=v0(m) +αv1(m, βac/α, t) , (12.4)

where

v0(m) =−m×heff[m] , (12.5)

and

v1(m, βac/α, t) =−m× (m×heff[m])+βacα

cos(ωt)m× (m× ep) . (12.6)

In equation (12.4), α has the role of a perturbation parameter and βac/α is a constant in the order of

unity. The unperturbed dynamics α=0 is the conservative precessional dynamics whose trajectories are

the contour lines of the free energy GL(m) on the unit-sphere. The phase portrait of the conservative

dynamics can be conveniently represented by its projection from the unit-sphere onto the (mx,my) plane

(see Figure 12.2).

mx

my

s2 s1mx=−1 mx= +1

d1

d2

u

Γ1Γ2

ΩcΩ2 Ω1Ω3

∆gc

Figure 12.2. Unit circle representation of conservative dynamics. Points s1, s2 are minima of the free energy GL.The point u is a maximum of GL . Points d1, d2 are saddles. Γ1,Γ2 are separatrix curves which separate low energyregions Ω1, Ω2, from the high energy region Ω3. Ωc is the chaotic region created close to the separatrices when theRF current exceeds a threshold value.


159

The phase portrait is composed by the regions Ω1,Ω2,Ω3 filled by a continuum of periodic trajectories.

The dynamics also admits equilibrium points which are minima of the free energy, like s1 and s2 associated

with mx= +1 and mx=−1 respectively, and maxima of the free energy like u1 and u2 (symmetric with

respect to the plane mz= 0 in the southern hemisphere, not visible in Figure 12.2). The regions Ω1,Ω2,

Ω3 enclosing the equilibria s1, s2, u1, u2 are separated by special magnetization trajectories Γ1,Γ2, referred

to as separatrices, which start and finish at the saddle points d1, d2.

When a small RF current is injected (βac ≪ 1 and α =/ 0), the dynamical system defined by (12.4)

becomes non-autonomous. It turns out that the combination between the time- varying forcing term and

the conservative magnetization motion close to the separatrices can produce strikingly complex dynamical

behavior. In fact, it can be shown [Ott02] that there is a threshold value of the RF current βac for which

a region Ωc of chaotic dynamics is created close to the separatrices (shaded region in Figure 12.2). This

phenomenon is referred to as homoclinic chaos [Ott02].

The homoclinic chaos in the region Ω1 can be detected by using the following Melnikov func-

tion [LLZ06, Per00]

M1(t0, βac/α) =

∫

−∞

+∞

mgd(t) · [v0(mgd(t))× v1(mgd(t), βac/α, t+ t0)] dt , (12.7)

where mgd(t) is the trajectory describing the unperturbed homoclinic trajectory Γ1. The zeros of the

Melnikov function reveal the onset of the chaotic dynamics. By substituting equations (12.6) and (12.6)

in equation (12.7), one obtains:

M1(t0, βac/α)=

∫

−∞

+∞

|mgd×heff[mgd]|2 dt

− βacα

∫

−∞

+∞

(mgd× ep) · (mgd×heff[mgd])cos(ω(t+ t0)) dt

, (12.8)

where the dependence of mgd on t is understood.

The central idea of the work is that the homoclinic chaos can participate in the microwave assisted

magnetization switching of the spin-valve. In fact, let us suppose that the magnetization is initially in

the equilibrium mx= +1. A DC field is applied along the x axis, with amplitude |hax| smaller than the

coercive field hc=Dy−Dx. In order to realize successful switching, magnetization has to overcome the

energy barrier defined by the anisotropy [BMS09], which is ∆gs(hax)= gd− gs1

∆gs(hax)=hc2

(

1− hax2

hc2

)

+hax

2

hc2

+ hax , (12.9)

where gd and gs1 are the values of the free energy associated with the saddle d and the equilibrium s1respectively (see Figure 12.2). Equation (12.9) implies ∆gs(0) =hc/2 and ∆gs(−hc)= 0.

Now we show how this energy barrier can be effectively reduced by exploiting the homoclinic chaos

produced by the injection of the RF current. The amplitude of the chaotic region Ωc can be found in

terms of a free energy variation ∆gc characterizing all the magnetization conservative trajectories enclosed

in Ωc. From Figure 12.2, one can capture the mechanism of RF assisted switching. In fact, as far as the


160

RF current βac increases, the region Ωc enlarges and increasingly overlaps the region Ω1 surrounding the

equilibrium s1. When Ωc completely fills Ω1, we conjecture that magnetization switching may occur.

15

10

5

0

−5

−0.23

−0.24

−0.25

−0.26

−0.27

−0.29 −0.28 −0.27 −0.26 −0.25 −0.24 −0.23 −0.22 −0.21 −0.2

0.16 0.18 0.2 0.22 0.24 0.26 0.28

hc′

ω

βac/α= 0.5

βac/α= 0.4

βac/α=0.3

βac/α= 0.2

ω= 0.25ω= 0.29

ω= 0.27

hax

×10−3

A

B

Figure 12.3. A) Energy barriers for the stable minimum mx =+1. The dashed line is obtained by equation (12.9).The solid lines represent the energy barrier reductions computed for βac/α =0.5 and frequency ω =0.25,0.27, 0.29.The values of parameters are: Dx =−0.26, Dy = 0.04, Dz = 0.94, α = 0.02. The coercive field is hc = Dy −Dx = 0.3.B) Reduced coercivity hc

′ computed from equation (12.12) as function of ω and βac/α.

In order to evaluate the amplitude ∆gc as function of hax and βac, we derive the balance equation for

the free energy GL. By multiplying both sides of equation (12.1) by heff[m], after some algebra, one has:

dGL

dt=−α

[

|m×heff[m]|2− βacα

cos(ωt)(m× ep) · (m×heff[m])

]

. (12.10)

It can be shown [BMS09] that ∆gc can be computed by using the Melnikov function (12.8):

∆gc⋍−αM1(t0, βac/α) , (12.11)

which gives the energy increase ∆gc with respect to the energy associated with the saddle as function of

βac and t0. We observe that ∆gc6 0 when βac= 0.


161

When βac =/ 0, in order to reduce the energy barrier between the equilibrium and the saddle, there

should exist an initial time t0⋆ such that the increase ∆gc should be positive. Thus, we compute the

effective reduction ∆gc⋆ of the energy barrier as ∆gc

⋆ = maxt0 ∆gc. Since both ∆gs and ∆gc⋆ depend on

hax, an estimation of the reduced coercivity hc′ achievable with the microwave current can be derived

from the equation:

∆gs= ∆gc⋆ . (12.12)

12.3 The numerical results.

In order to test the effectiveness of the proposed approach, we have considered a uniformly magnetized

anisotropic spin-valve with perpendicular polarizer ep = ez, effective anisotropy constants Dx = −0.26,

Dy = 0.04, Dz = 0.94 (here Dx takes into account both shape and uni-axial anisotropy along the x

direction). The coercivity, in absence of RF current (βac=0), is hc=Dy−Dx=0.3). In Figure 12.3A, the

behaviors of ∆gs and ∆gc⋆ as function of the DC field component hax are shown for a given βac/α= 0.5

and different values of frequency. The intersections between these curves (yellow dots in Figure 12.3A)

indicate the reduced coercivities hc′. By using (12.12), we have analyzed the dependence of hc

′ on ω and

βac/α. The results, reported in Figure 12.3B, show that RF assisted switching is possible in a moderately

wide frequency range ω ∈ (0.16, 0.29) (i.e. 4.5− 8.1 GHz if µ0Ms=1T) for RF current amplitudes above

the lower threshold βac= 0.2α. In addition, from Figure 12.3B, one can see that |hc′ | reduces as far as ωincreases at constant βac/α, and decreases as far as βac/α increases at constant ω.

ac ac ac ac ac

ac

Figure 12.4. (left) Dynamical hysteresis loops for different values of βac/α. Solid (dashed) lines refer to ω = 0.2(ω =0.25). (right) The figure reports a time trace of mx(t) for ω =0.2 and βac/α= 0.2. The values of parametersare the same as in Figure 12.3.


162

To test the accuracy of the theoretical predictions, we have performed numerical integration of the

LLS equation (12.1). In particular, the hysteresis loops in presence of the injected RF current have been

computed by slowly varying hax back and forth, in a linear fashion for a time interval T =2×105, in the

range (−0.4, 0.4). The results are reported in Figure 12.4 for several values of βac/α and ω = 0.2, 0.25

chosen according to the diagrams of Figure 12.3B. By comparing Figures 12.3B and 12.4, one can see

that the numerical simulations are in reasonable agreement with the theoretical predictions. We notice

that the theoretical estimate on the reduced coercivity is rather «pessimistic». In fact, it is apparent

from Figure 12.4 that the switching field can be reduced from 20% to more than 30% with respect to

the coercivity hc. Moreover, we observe that the switching process occurs in the sub-nanosecond regime

(100(γMs)−1∽0.57ns if µ0Ms=1T), as it can be inferred from Figure 12.4 where the time trace of mx(t)

is reported in the case of ω= 0.2 and βac/α= 0.2.

In conclusion, we remark that the proposed approach can be useful in the study of novel strategies

to achieve efficient microwave assisted magnetization switching.

12.3 The numerical results.

163

13Conclusions and Outlook

13.1 Conclusions

This thesis may be considered a summary of my current research interests, which lie within the areas of

Continuum Mechanics, Calculus of Variations and Partial Differential Equations. Indeed, the leitmotif of

the whole thesis is to apply ideas from these fields to variational problems related to the mathematical

study of magnetic phenomena that occur at vastly different spatial scales. In particular, most of the

results presented in this thesis have been motivated by problems arising from Micromagnetics.

Thus, the first part of the thesis (chapters 2 and 6) was mainly focused on the presentation of the

Micromagnetic Theory. The followed approach had the aim to present a clear and understandable link

between the classical theory of magnetic fields (due to Maxwell) and the micromagnetic theory: actually,

some effort was spent to highlight the role of constitutive relation played by Micromagnetic Theory in

the context of Maxwell’s theory of magnetic fields. The desire to achieve this end in a self-contained way,

was the main objective of chapter 2 in which a review of some fundamental magnetostatic concepts in

matter was presented.

Next, in the second part of the thesis (which may be defined as the union of chapters 3, 4 and 5) a new

and simple proof of the homogeneous ellipsoid problem relative to the Newtonian potential was presented.

The argument, essentially based upon the use of coarea formula, has permitted to reduce the problem to

the solution of the evolutionary eikonal equation, revealing in that way, the pure geometric nature of the

problem. Due to its physical relevance, particular attention has been paid to the three-dimensional case,

and in particular, to the computation of the demagnetizing factors which are one of the most important

quantities of ferromagnetism. Still, the desire to achieve this result in a self-contained way, was the

main objective of chapter 3, in which some results of classical potential theory were recalled. Finally, in

chapter 4 we focused on the introduction of the demagnetizing field which is one of the most important

and critical (being a long-range interaction energy term) quantity arising in micromagnetic theory.

The third part of the thesis (which may be defined as chapters 7 to 10) was, roughly speaking, devoted

to continue Brown’s work in the foundations of micromagnetic theory: indeed has Brown himself points

out in [Bro63]: «No claim is made that Micromagnetic theory has been fully developed; all that can be

said is that the foundations have been laid». In this respect, one of the main aims of the research activity

presented in this part was to gain a step further in the development of micromagnetic theory. More

precisely, chapter 8 has been devoted to the generalization of Brown’s fundamental theorem of the theory

of fine ferromagnetic particles to the case of a general ellipsoid [Fra11]. By the means of the best Poincaré

constant for the Sobolev space H1(Ω, R3), and some properties of the demagnetizing field operator, it

has been rigorously proven that for an ellipsoidal particle, with diameter d, there exists a critical size

(diameter) dc such that for d<dc the uniform magnetization states are the only global minimizers of the

Gibbs-Landau free energy functional GL. A lower bound for dc is then given in terms of the demagnetizing

factors.

165

Then, in chapter 9, Brown’s fundamental theorem for fine ferromagnetic particles has been extended to

the case of local minimizers of the Gibbs-Landau free energy functional on general ellipsoidal domains: it

has been proved that for an ellipsoidal particle, with diameter d, there exists a critical size (diameter) dcsuch that for d<dc the uniform magnetization states are the only local minimizers of the Gibbs-Landau

free energy functional GL. Moreover, it has also been proved that locally minimizing harmonic maps on

convex domains are constant in space.

Finally, in chapter 10, the homogenized functional of a periodic mixture of ferromagnetic materials

was derived: i.e. the Γ-limit of the Gibbs-Landau free energy functional, as the period over which the

heterogeneities are distributed inside the ferromagnetic body shrinks to zero.

The fourth and last part of the thesis (chapters 11 and 12) concerned the introduction of the

dynamic model (due to Landau Lifshitz) for the evolution of the magnetization inside a ferromagnetic

body. Indeed, nowadays, the challenging requirements of greater speed and areal density in magnetic

storage elements, ask for a comprehensive qualitative and quantitative understanding of nonlinear mag-

netization at nanometric scales. In particular, chapter 12, concerned the presentations of the results

exposed in [dDS+11] in which, by combining numerical and analytical techniques, was studied the role

of microwave spin-polarized electric current in assisting magnetization switching of an anisotropic mag-

netic particle. We have seen how the presence of a microwave spin-polarized current produces a reduction

of the coercive field associated with the particle anisotropy. In this respect, it has been analyzed how

this reduction is connected to chaotic magnetization dynamics which may be induced by the microwave

injected current. The presence of chaotic dynamics was predicted by an analytical technique based

on the Melnikov-Poincaré integral.

13.2 Outlook

Micromagnetic theory, and more generally continuum mechanics, provides a rich source of mathematical

problems and physical phenomena. In this respect, in the near (and long) future the attention will be

focused on three distinctive areas − characterized by different spatial scales − of the mathematical theory

of magnetism:

Stochastic homogenization. It is often the case that the microgeometries of heterogeneous spec-

imens are known only in a statistical sense. The analysis of composites with probabilistically defined

material properties plays a central role in many technology applications. The aim of the project is the

identification of the Γ-limit of the stochastic version of the Gibbs-Landau free energy functional in which

the material dependent parameters are now random variable. The mathematical tool-set will include

analytical methods from partial differential equations, multiscale analysis and the De Giorgi’s definition

of Γ-convergence of measure will be an interesting road to explore [dG84].

Stochastic Landau-Lifshitz-Gilbert (LLG)equation. The LLG equation is a strongly nonlinear

partial differential equation closely related to the nonlinear Schrodinger equation and to the so-called

heat flow of sphere-valued harmonic maps. Solutions of such a system of equations develop singularities

and vortex type solutions that are expected to lead to new types of magnetic memories operating in

nanoscales. In such scales, random fluctuations can modify behavior of solutions hence the LLG equations

must be modified to include random terms. The aim of the investigation would be to develop innovative

mathematical theory for solutions of stochastic partial differential equations that describe time evolution

Conclusions and Outlook

166

of magnetization in ferromagnetic materials. The mathematical tool-set will include analytical methods

from stochastic analysis, partial differential equations, multiscale analysis and the theory of dynamical

systems.

Figure 13.1. (left)Frog levitated in stable zone of a 16T magnet. (right)Levitation of a magnet 2.5m belowan unseen 11T superconducting solenoid stabilized by the diamagnetism of fingers (χ ≈10−5).

Newton-Coulomb potential. Motivated by the study performed in chapter 5, it would be inter-

esting the investigation of an extension of those results to different geometries. Shape plays a crucial

role in magnetostatics due to the long-range character of the interactions involved. These interactions

sense the finite extent of the magnetized body in a non-negligible way and cannot be reduced to an

effective additive or multiplicative energy rescaling [BDM09]. The analytical results concerning ellipsoidal

and rectangular-shaped bodies, have been used extensively to elucidate the quantitative aspects of the

geometry of domain structure. Moreover, very recently, highly non-trivial classes of inclusion shapes

have been discovered within the context of Eshelby’s elasticity problem [Liu08], which would exhibit

homogeneous strain states for homogeneous stresses. By analogy with the magnetostatic problem, the

same exotic shapes would support uniform magnetization [BDM09, Liu08]. The mathematical tool-set

will include analytical methods from potential theory and calculus of variations

Diamagnetic levitation. The ability of magnets to exerts forces on one another without touching

intrigues most children − and more than a few adults. It is a short step from pondering this curious

phenomenon to wondering whether the force from one magnet could be used to levitate another, seemingly

in defiance of gravity [Per04]. Unfortunately, a theorem due to Samuel Earnshaw proves that it is not

possible to achieve stable levitation (i.e. stable suspension of an object against gravity) of one magnet by

another with no energy input and this, roughly speaking, is mathematically due to the harmonic character

of r−1-type potentials involved which constraint the energy stationary points to be saddles rather than a

minimum: if the equilibrium is stable in one direction, it is unstable in an orthogonal direction. However,

the introduction of diamagnetic material at special spatial locations can stabilize such levitation [SHG01].

A magnet can even be stably suspended between (diamagnetic) fingertips and (see Figure 13.1), quite

recently, water droplets and even frogs have been levitated in this way at a magnetics laboratory in

the Netherlands [SG00]. Diamagnetic levitation is a striking physical phenomenon, one that has been

studied for many decades now [Per04]. Yet surprisingly few people, even scientists and engineers, are

familiar with it [Per04]. Even more surprisingly, no mathematical investigation of the subject has never

been made. In this respect, the aim of the project is to develop new mathematics for the identification of

energy configurations with prescribed local minimizers. The mathematical tool-set will include analytical

methods from potential theory and calculus of variations.

13.2 Outlook

167

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Index

Laplace’s second law . . . . . . . . . . . . . . . . . 14(CPP) geometry . . . . . . . . . . . . . . . . . . . 155Γ-convergence . . . . . . . . . . . . . . . . . . . . 134Γ-convergence of a family of functionals . . . . . . 139Γ-convergence of a sequence of functionals . . . . . 139almost-minimizers . . . . . . . . . . . . . . . . . . 140Ampère equivalence theorem . . . . . . . . . . . . . 17Ampère’s circuital law . . . . . . . . . . . . . . . . . 17Amperian currents . . . . . . . . . . . . . . . . . 18, 26anisotropy density energy . . . . . . . . . . . . 83, 137anisotropy energy . . . . . . . . . . . . . . . . 77, 135anisotropy energy density . . . . . . . . . . . . . . . 77anisotropy or crystalline-anisotropy or magneto-crystallineanisotropy energy . . . . . . . . . . . . . . . . . . . 70antisymmetric operator . . . . . . . . . . . . . . . . 13applied magnetic field . . . . . . . . . . . . . . . . 7, 67areal density . . . . . . . . . . . . . . . . . . . . . . . 9Biot-Savart law . . . . . . . . . . . . . . . . . . . 15, 55bit lines . . . . . . . . . . . . . . . . . . . . . . . . . 11Bohr magneton . . . . . . . . . . . . . . . . . . . . . 23Brown’s fundamental theorem of the theory of fine ferro-magnetic particles . . . . . . . . . . . . . . . . . . . 91Brown’s static condition for local equilibria . . . . . 86Brown’s static equations . . . . . . . . . . . . . . . . 87Cauchy’s integral formula . . . . . . . . . . . . . . . 55circulation free . . . . . . . . . . . . . . . . . . . . . 55classical solution . . . . . . . . . . . . . . . . . . . . 36Clausius form . . . . . . . . . . . . . . . . . . . . . . 73closed system . . . . . . . . . . . . . . . . . . . . . . 70configurational anisotropy . . . . . . . . . . . . . . 103conservative in Ω . . . . . . . . . . . . . . . . . . . . 58constitutive relation . . . . . . . . . . . . . . . . . . 20continuity condition . . . . . . . . . . . . . . . . . . 16continuous convergence . . . . . . . . . . . . . . . 140continuously convergent . . . . . . . . . . . . . . . 140convolution . . . . . . . . . . . . . . . . . . . . . . . 51crystalline anisotropy . . . . . . . . . . . . . . . . . 28cubic anisotropy . . . . . . . . . . . . . . . . . . . . 78Curie second law . . . . . . . . . . . . . . . . . . . . 34Curie temperature . . . . . . . . . . . . . . . . . . 7, 33curl free . . . . . . . . . . . . . . . . . . . . . . . . . 53damping constant . . . . . . . . . . . . . . . . . . 150demagnetizing factors . . . . . . . . . . . . . . . . . 19demagnetizing field . . . . . . . . . . . . . . . . . . 55demagnetizing tensor . . . . . . . . . . . . . . . . 153

diamagnetic substances . . . . . . . . . . . . . . . . 31dipole moment uε . . . . . . . . . . . . . . . . . . . 47Dirichlet problem . . . . . . . . . . . . . . . . . 35, 38distribution solution . . . . . . . . . . . . . . . . . . 36divergence . . . . . . . . . . . . . . . . . . . . . . . 36divergence free . . . . . . . . . . . . . . . . . . . . . 53divergence theorem . . . . . . . . . . . . . . . . . . 37domain structure . . . . . . . . . . . . . . . . . . . 135domain walls . . . . . . . . . . . . . . . . . . . . . 135domains . . . . . . . . . . . . . . . . . . . . . . 8, 20, 68double-layer potential . . . . . . . . . . . . . . . . . 40drift velocity . . . . . . . . . . . . . . . . . . . . . . 14E-two-scale converge . . . . . . . . . . . . . . . . . 141easy axes . . . . . . . . . . . . . . . . . . . . . . . 135easy axis anisotropy . . . . . . . . . . . . . . . . . . 78easy directions . . . . . . . . . . . . . . . . . . . . . 77easy plane anisotropy . . . . . . . . . . . . . . . . . 78effective demagnetizing tensor . . . . . . . . . . . 93, 94effective field . . . . . . . . . . . . . . . . . . . . . . 87elementary solution . . . . . . . . . . . . . . . . . . 37entropy . . . . . . . . . . . . . . . . . . . . . . . . . 70equicoercive . . . . . . . . . . . . . . . . . . . . . 139exchange energy . . . . . . . . . . . . . . . . . 94, 135exchange forces . . . . . . . . . . . . . . . . . . . . 7, 68exchange integral . . . . . . . . . . . . . . . . . . . . 79exchange interaction . . . . . . . . . . . . . . . 7, 21, 79exchange length . . . . . . . . . . . . . . . . . . . . 81exchange or exchange-stiffness energy . . . . . . . . . 70exchange parameter . . . . . . . . . . . . . . . . . 137exchange stiffness constant . . . . . . . . . . . . . . 81existence of a recovery sequence . . . . . . . . . . . 139exterior domains . . . . . . . . . . . . . . . . . . . . 41exterior scalar representation formula . . . . . . . . . 45ferromagnetic substances . . . . . . . . . . . . . . . 31fine particle . . . . . . . . . . . . . . . . . . . . . . 8, 67first law of thermodynamics . . . . . . . . . . . . . . 71flux free . . . . . . . . . . . . . . . . . . . . . . . . . 55forces due to spin-orbit coupling . . . . . . . . . . . 68free energy . . . . . . . . . . . . . . . . . . . . . . . 69free layer . . . . . . . . . . . . . . . . . . . . . . . . 10fundamental equations of magnetostatics in free space .17fundamental micromagnetic constraint . . . . . . . . 81fundamental solution . . . . . . . . . . . . . . . . . . 37Fundamental Theorem of Γ-convergence . . . . . . 140

173

fundamental theorem of vector analysis . . . . . . . 53Gauss’s law for magnetostatics . . . . . . . . . . . . 16generalized absolute temperature . . . . . . . . . . . 71generalized effective field . . . . . . . . . . . . . . 156generated heat . . . . . . . . . . . . . . . . . . . . . 71giant magneto-resistive effect . . . . . . . . . . . . . 10Gibbs state space . . . . . . . . . . . . . . . . . . . 70Gibbs-Landau free energy functional . . . . . . . 21, 75Gibbs-Landau free-energy density . . . . . . . . . . . 74Green’s first identity . . . . . . . . . . . . . . . . . . 38Green’s first identity for unbounded domains . . . . 41Green’s second identity . . . . . . . . . . . . . . . . 38Green’s second identity for unbounded domains . . . 41Green’s third identity for unbounded domains . . . . 43gyromagnetic ratio . . . . . . . . . . . . . . . . . . . 23hard axes . . . . . . . . . . . . . . . . . . . . . . . . 77harmonic function . . . . . . . . . . . . . . . . . . . 36Helmholtz free energy density . . . . . . . . . . . . . 74Helmholtz free energy functional . . . . . . . . . . . 75Helmholtz-Hodge decomposition formula . . . . . . . 53homoclinic chaos . . . . . . . . . . . . . . . . . . . 160Homogenized uni-axial anisotropy energy density . 148ideal dipole potential . . . . . . . . . . . . . . . . . 47ideal dipole potential distribution . . . . . . . . . . . 47ideal magnetic dipole of moment m . . . . . . . . . 18indivergent (or divergence free) . . . . . . . . . . . . 58integration by parts formula . . . . . . . . . . . . . . 37intensity of magnetization . . . . . . . . . . . . . . 7, 67interior scalar representation formula . . . . . . . . . 40internal energy . . . . . . . . . . . . . . . . . . . . . 70internal energy state function . . . . . . . . . . . . . 70internal potential energy . . . . . . . . . . . . . . . . 75internally entropy production . . . . . . . . . . . . . 72internally generated heat . . . . . . . . . . . . . . . 71irrotational (or curl free) . . . . . . . . . . . . . . . 58jump discontinuity . . . . . . . . . . . . . . . . . . . 48jump of the function u along ∂Ω . . . . . . . . . . . 46jump of the normal derivative . . . . . . . . . . . . . 46jump relations . . . . . . . . . . . . . . . . . . . 48, 49jump scalar representation formula . . . . . . . . . . 46jump scalar representation theorem . . . . . . . . . . 46laboratory frame . . . . . . . . . . . . . . . . . . . . 13Langevin function . . . . . . . . . . . . . . . . . . . 31Laplace operator . . . . . . . . . . . . . . . . . . . . 36Laplace’s equation . . . . . . . . . . . . . . . . . . . 36Laplace’s first formula . . . . . . . . . . . . . . . . . 15Laplacian . . . . . . . . . . . . . . . . . . . . . . . . 36Larmor angular velocity . . . . . . . . . . . . . . . . 30Larmor current . . . . . . . . . . . . . . . . . . . . . 30Larmor magnetic moment . . . . . . . . . . . . . . . 30Larmor precession . . . . . . . . . . . . . . . . . . . 30Lorentz local-field formula . . . . . . . . . . . . . . . 28Lorentz sphere method . . . . . . . . . . . . . . . . 27

Lorentz’s force law . . . . . . . . . . . . . . . . . . . 15Lyapunov structure . . . . . . . . . . . . . . . 151, 152magnetic constant . . . . . . . . . . . . . . . . . . . 16magnetic dipole moment associated to Ω . . . . . . . 17magnetic dipole-dipole forces . . . . . . . . . . . . . 68magnetic domains . . . . . . . . . . . . . . . . 20, 135magnetic doublet . . . . . . . . . . . . . . . . . . . . 18magnetic field . . . . . . . . . . . . . . . . . . . . . 26magnetic flux density field . . . . . . . . . . . . . 14, 55magnetic induction field . . . . . . . . . . . . . . . . 14magnetic local field . . . . . . . . . . . . . . . . . . 20magnetic or magnetostatic energy . . . . . . . . . . . 70magnetic polarization vector . . . . . . . . . . . . . 20magnetic poles . . . . . . . . . . . . . . . . . . . . . 17magnetic random access memories . . . . . . . . . . . 9magnetic susceptibility of the material . . . . . . 32, 32magnetic vector potential . . . . . . . . . . . . . . . 16magnetization . . . . . . . . . . . . . . . . . . . . . 20magnetization switching processes . . . . . . . . . 154magnetization vector . . . . . . . . . . . . . . . . . . 53magneto-crystalline energy . . . . . . . . . . . . . . 77magnetostatic scalar potential . . . . . . . . . . . 53, 55magnetostatic self-energy . . . . . . . . . . . . . . 135magnetostatic vector potential . . . . . . . . . . 53, 55magnetostrictive forces . . . . . . . . . . . . . . . . 68Mathematical Theory of Homogenization . . . . . . 134Maxwell’s fourth law . . . . . . . . . . . . . . . . . . 17medium-hard axes . . . . . . . . . . . . . . . . . . . 77Melnikov function . . . . . . . . . . . . . . . . . . 160Micromagnetic Theory . . . . . . . . . . . . . . . . . 20microscopic currents . . . . . . . . . . . . . . . . . . 19minimum demagnetizing factor . . . . . . . . . . . . 94molecular field . . . . . . . . . . . . . . . . . . . . . 29molecular field postulate . . . . . . . . . . . . . . . . . 7MRAMs . . . . . . . . . . . . . . . . . . . . . . . . . 9nearest point projection operator . . . . . . . . . . . 85Neumann boundary condition . . . . . . . . . . . . . 86Neumann problem . . . . . . . . . . . . . . . . . 35, 38Newton-Coulomb law . . . . . . . . . . . . . . . . . 55Newtonian kernel . . . . . . . . . . . . . . . . . . . . 37Newtonian potential . . . . . . . . . . . . . . . . . . 40non-dissipative . . . . . . . . . . . . . . . . . . . . 150normal to the boundary . . . . . . . . . . . . . . . . 58normalized anisotropy energy density . . . . . . . . . 81normalized anisotropy field . . . . . . . . . . . . . . 86normalized applied field . . . . . . . . . . . . . . . . 81normalized demagnetizing field . . . . . . . . . . . . 81normalized effective field . . . . . . . . . . . . . . . . 86open exterior of Ω . . . . . . . . . . . . . . . . . . . 41orbital current . . . . . . . . . . . . . . . . . . . . . 23orbital quantum number . . . . . . . . . . . . . . . . 23paramagnetic substances . . . . . . . . . . . . . . . 31Pauli exclusion principle . . . . . . . . . . . . . . . . 24

Index

174

permeability of free space . . . . . . . . . . . . . . . 16Philosophiae Naturalis Principia Mathematica . . . . 35physical dipole . . . . . . . . . . . . . . . . . . . . . 47pinned layer . . . . . . . . . . . . . . . . . . . . . . 10Plank constant . . . . . . . . . . . . . . . . . . . . . 23Poincaré constant . . . . . . . . . . . . . . . . . . . 95Poisson equation . . . . . . . . . . . . . . . . . . . . 52Poisson’s equation . . . . . . . . . . . . . . . . . . . 36potential . . . . . . . . . . . . . . . . . . . . . . . . 35potential field . . . . . . . . . . . . . . . . . . . . . 35potential function . . . . . . . . . . . . . . . . . . 35Q-periodic . . . . . . . . . . . . . . . . . . . . . . 137Quantized projection of the orbital angular momentum .24Robin problem . . . . . . . . . . . . . . . . . . . . . 35saturation magnetization . . . . . . . . . . . . . . 134saturation value of the magnetization . . . . . . . . . 33second law of thermodynamics . . . . . . . . . . . . 73simple-layer potential . . . . . . . . . . . . . . . . . 40solenoidal in Ω . . . . . . . . . . . . . . . . . . . . . 58spin momentum . . . . . . . . . . . . . . . . . . . . 24spin valve . . . . . . . . . . . . . . . . . . . . . . . . 10spin-polarized currents . . . . . . . . . . . . . . . . . 11spontaneous magnetization . . . . . . . . . . . . . . 81spontaneous magnetization state . . . . . . . . . . . . 9Stokes identity . . . . . . . . . . . . . . . . . . . . . 39

surface potentials . . . . . . . . . . . . . . . . . . . 40tangent to the boundary . . . . . . . . . . . . . . . . 58tangentially homogenized energy density . . . . 142, 143telegraphone . . . . . . . . . . . . . . . . . . . . . . . 9Tesla . . . . . . . . . . . . . . . . . . . . . . . . . . 15thermodynamic equilibrium states . . . . . . . . . . 70third Green’s identity . . . . . . . . . . . . . . . . . 39total atomic magnetic moment . . . . . . . . . . . . 24transmission or (jump) conditions . . . . . . . . . . 56two-scale convergence . . . . . . . . . . . . . . . . 134Two-scale convergence . . . . . . . . . . . . . . . . 141uni-axial anisotropy . . . . . . . . . . . . . . . . . . 78vector of uncompensated heats . . . . . . . . . . . . 70vector of work variables . . . . . . . . . . . . . . . . 70volume and surface source charge distributions . . . 55volume and surface vortex current distributions . . . 55weak Brown’s static equation . . . . . . . . . . . . . 86weak Euler-Lagrange equation . . . . . . . . . . 86, 87Weber . . . . . . . . . . . . . . . . . . . . . . . . . . 15Weiss constant . . . . . . . . . . . . . . . . . . . . . 29Weiss domains . . . . . . . . . . . . . . . . . . . . . 29Weiss molecular field theory . . . . . . . . . . . . . . 20word lines . . . . . . . . . . . . . . . . . . . . . . . . 11work performed on the system . . . . . . . . . . . . 71Zeeman energy . . . . . . . . . . . . . . . . . . . . 135

Index

175

On the nature of magnetization states minimizing the micromagnetic free energy functional.

Documents

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lorentz local field

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