A Sharp-Interface Limit for a Two-Well Problem in Geometrically Linear Elasticity

A sharp-interface limit for a two-well problem in

geometrically linear elasticity

Sergio Conti1 and Ben Schweizer2

April 21, 2005

Abstract: In the theory of solid-solid phase transitions thedeformation u : Ω ⊂ R

d → Rd of an elastic body is determined

via a functional containing a nonconvex energy density and asingular perturbation. We study

Iε[u] =

∫

Ω

1

εW (∇u) + ε|∇2u|2 .

Frame indifference, within a linearized setting, requires thatW depends only on the symmetric part of ∇u. The potentialW is non-negative and vanishes on two wells, i.e., for d = 2,on two lines in the space of matrices.We determine, for d = 2, the Gamma limit I0 = Γ− limε→0 Iε.The limit I0[u] is finite only for deformations u that fulfillW (∇u) = 0 almost everywhere and have sharp interfaceswhere ∇u has jumps. For these u, I0[u] equals the line in-tegral over the interfaces of a surface energy density.

1 Introduction

Modeling of phase transitions in solids leads to functionals of the form

Eε[u,Ω] =

∫

Ω

W (∇u) + ε2|∇2u|2 , (1.1)

where u : Rd ⊃ Ω → R

d is the elastic displacement, W a free energy densitywith multiple minima, and ε > 0 a small parameter. The free energy prefersdeformations u with ∇u near the set of minima of W , the second part of theenergy penalizes transitions from one gradient to another. The parameter εdetermines the width of domain walls [5, 19, 24, 7].

Variational problems as in (1.1) have often been proposed both for numer-ical and analytical computations, but the presence of different length scales

1Fachbereich Mathematik, Universitat Duisburg-Essen, Campus Duisburg, Lotharstr.65, D-47057 Duisburg, Germany

2Mathematisches Institut, Universitat Basel, Rheinsprung 21, CH-4051 Basel, Schweiz

1

renders the treatment difficult. At a heuristic level, various simplified formshave been used in which the singular perturbation is replaced by a measureof the length of the interface. The connection between the two formulationshowever has not been clarified so far.

The method of choice for the study of the asymptotic behaviour of vari-ational problems is Gamma convergence, as developed by De Giorgi and hisschool in the 70s ([17]; see also [12, 10, 9]). The first application of Gammaconvergence was obtained by Modica and Mortola [23], who studied the scalarproblem

Jε [v,Ω] =

∫

Ω

1

εW (v) + ε|∇v|2 , (1.2)

where W (v) = 0 iff v ∈ a, b, which arises in the van der Waals–Cahn-Hilliard theory of fluid-fluid phase transitions. They have shown that forsome k = k(W ) > 0, the Gamma-limit is Γ − limε→0+ Jε = J0,

J0 [v; Ω] =

k PerΩ(E) if v = χEa+ (1 − χE)b ∈ BV (Ω; a, b) ,+∞ else.

(1.3)

This shows that, in the case of fluid-fluid phase transitions, the limitingproblem corresponds to minimizing the area of the interface.

Generalizations of (1.2)-(1.3) were obtained by Bouchitte [8] and by Owenand Sternberg [25] for the coupled problem, in which the integrand in Jεhas the form ε−1f(x, v(x), ε∇v(x)). We also refer to the work of Kohn andSternberg [20] where the study of local minimizers for (1.2) was undertaken.The vector-valued setting was considered in [16, 6]. The case where W hasmore than two wells was addressed by Baldo [3] (see also Sternberg [26]),and later generalized by Ambrosio [1].

The inclusion of functionals of elastic problems, such as (1.1), into thisframework has defied a considerable mathematical effort during the pastdecade. In general, one would like to understand the Gamma limit of

Iε[u,Ω] =

∫

Ω

1

εW (∇u) + ε|∇2u|2 , (1.4)

where u : Rd ⊃ Ω → R

d stands for the deformation, and, taking into accountframe-indifference, the free energy density W (F ) vanishes for F ∈ SO(d)A∪SO(d)B, where SO(d) is the set of rotations in R

d. In order to guarantee theexistence of continuous, non-affine weak solutions for the limiting problemthe two wells must be rank-one connected (this is often called Hadamard’scompatibility condition for layered deformations, see Ball and James [5]),in the sense that there must be rotations Q, Q′ such that QA − Q′B is arank-one matrix, say a ⊗ ν. Finite energy deformations u of the limiting

2

problem are then necessarily piecewise affine and interfaces are planes. Morespecifically, a region with ∇u = QA is separated by one with ∇u = Q′B bya hypersurface with normal ν.

This shows that the limiting problem is much more rigid than in the caseof fluid-fluid phase transition – in accordance with experimental observationsof very specific laminar structures in shape-memory alloys. At first glancethe analysis may seem to be greatly simplified as compared with the problem(1.2) which requires handling minimal surfaces, and one is tempted to definev = ∇u and apply the same methods as above. However, it turns out thatthe PDE constraint v = ∇u (or, equivalently, curl v = 0) imposed on theadmissible fields presents numerous difficulties to the characterization of theΓ-limsup. Precisely, the main obstacle in the proof is as follows. Given ∇uwith a layered structure with two interfaces and three affine regions, it ispossible to construct a recovery sequence (a sequence realizing the optimalenergy) near each interface, starting from a generic optimal sequence in theΓ-liminf. But the task of gluing together the two sequences on a suitablelow-energy intermediate layer is very delicate. It can be done only if anadditional structure of low-energy sequences can be exploited, which in turncan be obtained by using suitable rigidity properties.

A first simplification of the problem, in which the frame-indifference con-straint was completely neglected and replaced by the assumption W (F ) = 0iff F ∈ A,B, with A − B = a ⊗ ν, was recently studied in [11]. Thiswork was based on a two-step interpolation in the construction of realizingsequences, which exploited the fact that no rotations of the gradient are al-lowed. An intermediate case between (1.2) and (1.4), where the nonconvexpotential depends on u and the singular perturbation on ∇2u, was studiedby Fonseca and Mantegazza [15]. If u is a scalar field on a two-dimensionaldomain, and W vanishes on the unit circle, e.g. W (∇u) = (1−|∇u|2)2, then(1.4) reduces to the Eikonal functional which arises in the study of liquidcrystals and in of blistering of delaminated thin films. The Eikonal problemhas received considerable mathematical attention, but in spite of substantialprogress (see [2, 18, 13, 22]) its Gamma limit remains to be identified.

In this work, we consider the problem (1.4) in two dimensions, includ-ing the requirement of rotational invariance within a geometrically linearframework. This means that we replace invariance under SO(2) by invari-ance under its tangent space, i.e. by invariance under the additive action ofantisymmetric matrices

Rϕ =

(

0 −ϕϕ 0

)

, W (F ) = W (Rϕ + F ) for all ϕ ∈ R. (1.5)

The linear structure of this condition makes many computations easier, and

3

is indeed very often used when explicit numerical results are desired (seee.g. [7]). Some abstract results are however more complex for the linearizedproblem, due to the loss of compactness of the zero-set of W . In the presentwork the main advantage of using the geometrically linear theory is in thedecoupling of the angular part in computing volume averages in Section 4.2.

We consider energies W that vanish on two matrices A,B ∈ R2×2. By

frame indifference, W then vanishes on two orbits, i.e. W (F ) = 0 if and onlyif, for some ϕ ∈ R, F = Rϕ + A or F = Rϕ + B. Further, we assume Wto be continuous and to have quadratic growth, both at infinity and close tothe wells. We will show that the functionals Iε Gamma-converge to

I0[u,Ω] =

∫

J∇usymk(ν)dH1 if ∇usym = 1

2(∇u+ ∇uT ) ∈ BV (Ω, A,B)

+∞ else,

(1.6)where J∇usym denotes the jump set of ∇usym, and ν the normal to it.

Theorem 1.1. Let Iε be as in (2.1) with M as in (2.4), W : R2×2 →

R a nonnegative function which is invariant under linarized rotations andvanishing on two symmetric matrices A and B as in (2.2–2.3) with det(A−B) < 0, and I0 as in (1.6) with k of Definition 1.2. Then, for any open,bounded, strictly star-shaped domain Ω ⊂ R

2, we have

Γ − limε→0

Iε = I0

with respect to the strong W 1,2 topology.

We say that an open set Ω is strictly star-shaped if there is a point z ∈ Ω,such that for any z′ ∈ ∂Ω the open segment (z, z′) is contained in Ω.

Proof. The lower bound is obtained in Proposition 3.1, the realizing sequenceof the Γ−lim sup is constructed in Proposition 5.1. By the compactness resultof Theorem 2.1 the L1 and the W 1,2 topologies give the same Γ-limit.

The main restriction of the above theorem is the limitation to two dimen-sions. While we are not aware of any reason why the natural generalizationof Theorem 1.1 to n > 2 should not hold, it seems that the geometric ar-guments leading to the rigidity estimates of section 4 can not be extendedeasily to higher dimensions.

The rest of the paper is organized as follows. In Section 2 we stateour assumptions, reduce the problem to a canonical form, and show thecompactness of finite-energy sequences in W 1,2(Ω). Section 3 contains thelower bound. It uses the compactness result, scaling arguments, and the

4

implicit characterization of k(ν) in Definition 1.2. Section 4 introduces thenew techniques to derive rigidity estimates (see also the discussion below).In Section 5 the rigidity result is used to construct the realizing sequencesthat yield the upper bound for the Γ-limit. Section 6 provides two examples.The first one shows that the surface energy k(ν) of Definition 1.2 cannotbe, in general, determined by minimizing over one-dimensional profiles. Thesecond one shows that the H1/2 rigidity estimate in Section 4 is optimal, inthe sense that a corresponding estimate in H1 does not hold.

One main new ingredient of the present proof is the rigidity estimateof Section 4. Roughly speaking, for a low energy sequence uε, the energycontrols the distance of ∇uε from the wells A+Rϕ, ϕ ∈ R and B+Rϕ, ϕ ∈R. The energy does not, however, control the variable ϕ ∈ R; this ’angle’must be controlled using the fact that we are dealing with a gradient field.This is analogous to Korn’s inequality, which states that the symmetric partof the gradient controls, up to an additive constant, the full gradient (in Lp).The corresponding nonlinear result is the classical Liouville rigidity, whichstates that a gradient field taking values in the set of rotations SO(d) isconstant (on connected domains), or equivalently that locally isometric mapsfrom R

d to Rd are affine. In this sense, our rigidity result can be regarded as a

Korn inequality for two wells. Our rigidity estimate is derived by introducinga new method based on a finite-element grid that refines towards a line. Itpermits to show that for every low energy deformation uε there are manylines on which uε is close to an affine function, in the H1/2 norm.

In closing, we state the definition of the surface energy k(ν) and introducesome notation.

Definition 1.2. Let ν ∈ R2 be a unit vector such that, for some antisymmet-

ric matrix S ∈ R2×2 and some vector a ∈ R

2 there holds A = B + S + a⊗ ν.Let furthermore Qν be the unit square centered in the origin with one sidenormal to ν, and let uν0 : Qν → R

2 be the Lipschitz-continuous function thatvanishes at the origin and with ∇uν0(z) = A for z ·ν > 0, and ∇uν0(z) = B+Sfor z · ν < 0. We set

k(ν) = inf

lim inf Iεi[ui, Qν ] : εi → 0, ui → uν0 in L1

.

Notation. We use the letter c for constants that do not depend on ε andu, but only on W and M ; the value may change from line to line. We writez = (x, y) for points in R

2, with x, y ∈ R. Correspondingly ex, ey ∈ R2

is the canonical basis of R2, and ux, uy the two components of a function

u : Ω → R2.

5

2 Preliminaries and compactness

We consider, for ε > 0, Ω a bounded, open, Lipschitz subset of R2, and

u : Ω → R2, the functionals

Iε[u,Ω] =

∫

Ω1εW (∇u) + ε〈M∇2u,∇2u〉 if u ∈W 2,2(Ω)

∞ else.(2.1)

Here, W is an energy density of a two-well problem respecting the frameindifference in a geometrically linear setting, i.e.,

W : R2×2 → R , continuous, W ≥ 0 , W (F ) = W (F sym) (2.2)

where F sym = (F + F T )/2 denotes the symmetric part of F , and there aretwo symmetric matrices A and B with det(A−B) < 0 (the meaning of thiscondition is discussed below) such that

W (F ) = 0 iff F sym ∈ A,B.

We further assume that W has quadratic growth, both close to the minimaand at infinity, in the sense that there are constants c and c′ such that

cW0(F ) ≤ W (F ) ≤ c′W0(F ) , (2.3)

where W0 is the squared distance from the two wells,

W0(F ) = min(

|F sym − A|2 , |F sym −B|2)

.

The singular perturbation has been here generalized to an elliptic quadraticform characterized by a symmetric linear map M such that

M : R2×2×2 → R

2×2×2 , 〈MF,F 〉 > 0 for all F such that Fijk = Fikj,

which means that there are c, c′ such that

c|∇2u|2 ≤ 〈M∇2u,∇2u〉 ≤ c′|∇2u|2 (2.4)

for any smooth map u : Ω → R2.

Heuristically, one expects the gradient of the limit function to take onlyvalues whose symmetric part equals A and B. Nontrivial maps of this kindare possible only if the two wells are rank-one connected. To see this, weconsider an interface between a region where ∇u0 = A + S1 and one where∇u0 = B + S2, with S1 and S2 antisymmetric (i.e. Ssym

1 = Ssym2 = 0). Then,

if enough regularity is present to define a tangent to the interface and traces

6

on both sides, the tangential parts of the gradient must coincide. Hencethe difference of the two gradients (A + S1) − (B + S2) must be a rank-onematrix a ⊗ ν, where ν is the normal to the interface. In particular, we seekan antisymmetric S = s(ex ⊗ ey − ey ⊗ ex) such that

A = B + S + a⊗ ν . (2.5)

The fact that A−B − S is rank one implies that

0 = det(A−B − S) = det(A−B) + s2 .

Therefore (2.5) has solutions only if det(A−B) ≤ 0. We shall focus here onthe generic case det(A−B) < 0, where (2.5) has two distinct solutions.

The problem can be reduced to a canonical form via an affine change ofvariables. Growth conditions and star-shapedness of the domain are clearlypreserved. Invariance of W under addition of antisymmetric matrices is alsopreserved, if we consider u(z) = P T u(Pz) + Qz, with an invertible matrixP and arbitrary matrix Q. Indeed, the deformation gradient F transformsaccording to F = P T FP + Q, and P TSP is antisymmetric for S antisym-metric by (P TSP )T = P TSTP = −P TSP . Setting W (F ) = W (F ), the firstterm of the energy density is unchanged. For the second gradient we obtain

∇2u = P∇2uP T ⊗ P T ,

inducing onM the affine change Mijk lmn = Mi′j′k′ l′m′n′Pii′PTjj′P

Tkk′Pll′P

Tmm′P T

nn′ .This leaves the ellipticity condition (2.4) unaffected. Finally, with appropri-ate matrices P and Q we can reduce to the case

A = 0 and B = ex ⊗ ey + ey ⊗ ex (2.6)

To see this, it suffices to set Q = A, so that A = 0 for any choice of P , andthen select P which brings B = B − A into the form above. The latter canbe done in two steps: by the representation theorem for quadratic forms weachieve the form B = ex⊗ex−ey⊗ey, and, by another transformation, (2.6).

We turn to the compactness result. It exploits a combination of thearguments used e.g. in [16, 11] for the case that W vanishes on a finite set,and the additional rigidity which comes from Korn’s inequality. We recallthat Korn’s inequality states that for all maps u ∈ W 1,2(Ω,R2), where Ω isa bounded set in R

2 with Lipschitz boundary, there is ϕ ∈ R such that

∫

Ω

∣

∣

∣

∣

∇u−(

0 −ϕϕ 0

)∣

∣

∣

∣

2

≤ cΩ

∫

Ω

|∇u+ (∇uT )|2 , (2.7)

where cΩ depends only on Ω (see e.g. Theorem 62.F in [27]).

7

Theorem 2.1 (Compactness). Let ui, εi be sequences such that εi → 0and Iεi

[ui,Ω] ≤ C <∞, and let assumptions (2.1-2.4) hold. Then there is asubsequence of ui, and sequences ai, bi, ϕi ∈ R such that

vi(x, y) = ui(x, y) −(

aibi

)

+

(

0 −ϕiϕi 0

)(

xy

)

converges strongly in W 1,2 to u0, with ∇usym0 ∈ BV (Ω, A,B).

Proof. We first observe that∫

Ω

|∇usymi |2 ≤ c

∫

Ω

[W (∇ui) + 1] ≤ cIεi[ui,Ω] + c|Ω|

is uniformly bounded. By Korn’s inequality (2.7) the same is true for thefull gradient of ui, after subtracting an antisymmetric linear map. Thereforewe can choose ai, bi, ϕ

′i and a subsequence such that (after relabeling)

vi = ui −(

ai − ϕ′iy

bi + ϕ′ix

)

u0 weakly in W 1,2 . (2.8)

We now show that the symmetric part of the gradient of vi converges strongly,using Young measures [4, 24]. Let fi = ∇vsym

i . Since it has a weak limit inL2, it generates a Young measure νzz∈Ω. Since W (∇ui) = W (fi), we getthat

∫

W (fi) → 0, and hence

0 = limi→∞

∫

Ω

W (fi) ≥∫

Ω

∫

R2×2

W (ξ)dνz(ξ) dz .

This shows that the Young measure is supported on the set of symmetricmatrices contained in the null set of W ,

νz = (1 − θ(z))δA + θ(z)δB for a.e. z ∈ Ω . (2.9)

We introduce the geodesic distance dW (F,G) induced by the potential W ,

dW (F,G) = inf

∫ 1

0

√

W (g(s))|g′(s)| ds , (2.10)

where the infimum is taken over piecewise C1-functions g : [0, 1] → R2×2 with

g(0) = F and g(1) = G. It is clear that dW (F,A) = 0 iff F sym = A, and thesame for B. On the other hand, dW (A,B) > 0. We claim that dW (fi(z), A)is uniformly bounded in W 1,1 (and hence has a subsequence that convergesweakly in BV). To see this, we compute

∫

Ω

|∇dW (fi(z), A)| ≤∫

Ω

√

W (fi) |∇fi| ≤ cIε[ui,Ω] ≤ C ′

8

and exploit the quadratic growth of W for the L1 estimate (for the com-pactness argument, the growth requirements can be relaxed via a standardtruncation argument, see e.g. [16, 11]. Note however, that Korn’s inequalitydoes not hold in W 1,1, hence p-growth from below with p > 1 is required inthis case). Therefore, dW (fi(z), A) has a subsequence which converges weaklyin BV and strongly in L1 to some BV function g. This implies that the cor-responding Young measure µ is a Dirac mass almost everywhere. Equation(2.9) yields, however, µz = (1 − θ(z))δdW (A,A) + θ(z)δdW (B,A) for a.e. z ∈ Ω.We conclude that θ(z) ∈ 0, 1 a.e., i.e. that fi converges strongly in L2.

By the uniqueness of the weak limit and (2.8) it is also clear that fi →∇usym

0 . Let us consider the sequence wi = vi−u0. By the previous argumentsthe symmetric part of the gradient converges to zero strongly in L2, and witha further application of Korn’s inequality we obtain that

∫

Ω

∣

∣

∣

∣

∇wi −(

0 −ϕ′′i

ϕ′′i 0

)∣

∣

∣

∣

2

≤ cΩ

∫

Ω

|∇wsymi |2 → 0 ,

for some sequence ϕ′′i ∈ R. The result follows, with ϕi = ϕ′

i + ϕ′′i .

The next statement characterizes the structure of possible limits u offinite-energy sequences. A corresponding result for the geometrically nonlin-ear case was obtained by Dolzmann and Muller [14]. See Figure 3.1 belowfor a graphical illustration.

Proposition 2.2. Let u ∈ W 1,2 satisfy ∇usym ∈ BV (Ω, A,B). Then ∇uis constant in each connected component of Ω \ J , where J = J∇usym is thejump set of ∇usym, and the normal to J can take only the values ±ν1 and±ν2, as given by Eq. (2.5). The set J is the union of countably many disjointsegments, normal either to ν1 or to ν2 whose endpoints belong to ∂Ω.

Proof. We can assume that (2.6) holds. Consider a square Q = (x0, y0) +(0, δ)2 contained in Ω. Since Q is convex and u = (ux, uy) satisfies ∂xux =∂yuy = 0 a.e. in Q,

ux = ux(y) , uy = uy(x) .

The symmetric part of the gradient becomes

∇usym(x, y) =[

u′x(y) + u′y(x)] ex ⊗ ey + ey ⊗ ex

2

whence (u′x(y) + u′y(x))/2 is a BV function which takes values 0 and 1. Forsuch a function, necessarily one of the functions u′x, u

′y is constant, the other

9

takes only two values (see Lemma 2.3 below). Therefore J ∩ Q is the finiteunion of either horizontal or vertical segments whose endpoints are in ∂Q.

This argument can be applied to any square contained in Ω, hence Jis composed by disjoint horizontal and vertical segments whose endpointsbelong to ∂Ω, and Ω \ J is open. The function ∇u is constant in eachconnected component of Ω \ J , and, since H1(J) <∞, J consists of at mostcountably many segments.

In the above proof we used the following lemma, which follows immedi-ately by considering pairs of points in (0, 1)2, which are Lebesgue-points ofall involved functions.

Lemma 2.3. Let f , g ∈ L1((0, 1),R), and assume that

f(x) + g(y) ∈ 0, 1 for a.e. (x, y) ∈ (0, 1)2.

Then, either f or g is a constant function in the sense of L1((0, 1),R).

3 Lower bound

In this section we show that the limiting energy of sequences uε is boundedfrom below by the functional I0 of (1.6).

Proposition 3.1 (Lower bound). Let Ω be an open, bounded, Lipschitzdomain, and assume that (2.1-2.4) hold. Then, for all u0 ∈ L1(Ω) and allsequences ε→ 0 and uε → u0 in L1, we have

lim infε→0

Iε[uε,Ω] ≥ I0[u0,Ω] .

The argument for the lower bound is similar to the one used in the caseof W vanishing for two matrices in [11], and is composed of two main ingre-dients. First, we characterize the lower bound for rectangular domains whichcontain a single interface. Then, using the compactness and structure resultsof Section 2, we show that if the liminf is finite, then an arbitrarily largefraction of the limiting energy is contained in a finite union of disjoint rect-angles, each of which contains a single interface, see Figure 3.1. We assumethat (2.6) holds, and let Sh and Sv be antisymmetric matrices such that

(B + Sh) − A = ah ⊗ νh , (B + Sv) − A = av ⊗ νv , (3.1)

with νh = ey and νv = ex. The index indicates that the first are horizontalinterfaces, the latter vertical ones.

10

Figure 3.1: Example of a possible limit function with three interfaces. Thethree rectangular boxes capture most of the energy. The letters denote thevalue of the symmetric part of ∇u0 in the different regions.

We consider a rectangular domain (−d, d) × (−l, l) and a limit functionwhich contains a single horizontal interface in the center,

u±h (x, y) = S

(

xy

)

+

A(x, y)T if ± y < 0

(B + Sh)(x, y)T if ± y ≥ 0 ,

(3.2)

where S is any antisymmetric matrix, and Sh is as in (3.1). The optimalenergy needed to approximate this interface is

F±h (d, l, S) = inf

lim infi→∞

Iεi[ui, (−d, d) × (−l, l)] : εi → 0, ui → u±h in L1

.

We shall show that the limiting energy per unit interfacial length does notdepend on the orientation of the interface, on the domain size, and on thesuperimposed rigid rotation, in the sense that F±

h (d, l, S) = 2dkh for somekh depending only on W and M .

Analogously, for vertical interfaces we define F±v (d, l, S) using, on the

domain (−l, l) × (−d, d), the limiting functions

u±v (x, y) = S

(

xy

)

+

A(x, y)T if ± x < 0

(B + Sv)(x, y)T if ± x ≥ 0,

where Sv is as in (3.1), and S is any antisymmetric matrix. Note that we used for lenghts along the interface and l for lengths in the orthogonal direction.

Lemma 3.2. There are constants kh, kv depending only on W and M suchthat F±

h (d, l, S) = 2dkh, F±v (d, l, S) = 2dkv.

11

Proof. Since W (F + S) = W (F ), and ∇2(u + S(x, y)T ) = ∇2u, it is clearthat the result does not depend on S, which can therefore be dropped fromthe notation. Furthermore, the result does not depend on the orientationof the interface, i.e., F+ = F−. Indeed, the energy is invariant under theoperation u→ Tu defined by

(Tu)(z) = −u(−z) .Indeed, (∇Tu)(z) = (∇u)(−z), hence the integral of W (∇u) is un-

changed. The second term is also unchanged since it is even and (∇2Tu)(z) =−(∇2u)(−z). On the other hand, if uj → u+

h , then Tuj → Tu+h = u−h , and

vice versa. Therefore F+h (d, l) = F−

h (d, l), hence we can focus on the firstone and drop the superscript.

It remains to consider the dependence on d and l. By restricting theintegration we see that Fh(d, l) is nondecreasing in l. Considering sequencesvi(z) = αui(z/α) and αεi we find

Fh(αd, αl) = αFh(d, l)

for any α > 0. By dividing the domain in n translated copies of (−d/n, d/n)×(−l, l), and restricting u to the one where the energy is lowest, we get

Fh

(

1

nd, l

)

≤ 1

nFh(d, l)

for n ∈ N. Now,

1

nFh (d, l) = Fh

(

1

nd,

1

nl

)

≤ Fh

(

1

nd, l

)

≤ 1

nFh (d, l) ,

hence equality must hold throughout, and in particular, using the monotonic-ity in l, Fh(d, l) does not depend on its second argument. The scaling derivedabove gives then the result. The same argument works for Fv(d, l).

We are now ready to prove the main result of this Section.

Proof of Prop. 3.1. If the lim inf is infinite there is nothing to prove. Oth-erwise, using the compactness result (Theorem 2.1) we obtain that the limithas the structure given by Proposition 2.2, and that I0 is finite. In the fol-lowing we only need to consider such limits. Further, it is sufficient to showthat for any δ > 0 the inequality holds up to an error term controlled by δ.

The jump set of ∇usym0 is composed by a countable union of segments,

which are either normal to νh or normal to νv. We denote it by

J(∇usym0 ) =

∞⋃

i=1

Ihi × yi ∪∞⋃

i=1

xi × Ivi

12

(one or both unions can be finite, in which case the next step is not needed).For each δ, there is N such that the first N segments of each of the sumscover at least a 1 − δ fraction of the total measure, i.e.,

N∑

i=1

|Iαi | ≥ (1 − δ)∞∑

i=1

|Iαi |

for α ∈ h, v. For each of those intervals Iαi , consider a compactly containedsubinterval Jαi of length |Jαi | ≥ (1 − δ)|Iαi |. We claim that there is h > 0such that the rectangles

ωhi = Jhh × (yi − h, yi + h) , ωvi = (xi − h, xi + h) × Jvh

for 1 ≤ i ≤ N are contained in Ω, disjoint, and each contains a singleinterface. To see this, we consider the sets

K =N⋃

i=1

Jhi × yi ∪N⋃

i=1

xi × Jvi , H =∞⋃

i=N+1

Ihi × yi ∪∞⋃

i=N+1

xi × Ivi ,

and show that their closures, K and H, are disjoint. This follows from thefact that interfaces can only meet in their end-points which belong to ∂Ω,that K does not contain any point of ∂Ω, and that H has finite length,whence cluster points of H are necessarily in H ∪ ∂Ω. By compactness, Kand H ∪ ∂Ω have a positive distance, and it suffices to take h less than thisdistance to prove the claim.

It follows that the ωhi are of the kind considered in the definition of Fh

and the ωvi of the kind considered in the definition of Fv, and therefore

lim infε→0

Iε[uε,Ω] ≥∑

i,α

lim infε→0

Iε[uε, ωαi ] ≥

N∑

i=1

kh|Jhi | +N∑

i=1

kv|Jvi |

≥ (1 − δ)2

∞∑

i=1

[

kh|Ihi | + kv|Ivi |]

= (1 − δ)2I0[u0,Ω] .

Since this can be done for any δ, the proof is complete.

4 Rigidity

In this section we show that, given a small-energy deformation u, there aremany cross-sections Σ of the domain such that the restriction of u is close toan affine function in the H1/2(Σ) norm. This makes quantitative the fact that

13

if a gradient field is pointwise close to either A+Rϕ or B+Rϕ, and interfacesbetween the A and B regions are short, then it is close to a constant map.The estimate will be used in the next section to obtain the upper bound.

If only one of the two phases were present, at least on a side of theconsidered cross-section, then the H1/2 estimate on a line would follow forevery line immediately from Korn’s inequality and a trace theorem. Thisis, however, not true for generic small-energy functions, as we show with anexample in Remark 6.1. We instead will show that u can be replaced onone side of an appropriately chosen line by a piecewise linear function whichachieves a similar elastic energy by using only one of the two phases. Theargument is based on compatibility conditions on a self-similar grid.

The precise strategy is as follows. In Section 4.1 we show that for manyvalues of y0, both the energy along the horizontal line at height y0, and theaverage energy in the stripes (y0−2−k, y0) are controlled by the global energy.We then construct a grid that refines towards one of those good horizontallines, such that on each edge of the grid the energy is small. This impliesthat only one phase is used on the edges of the grid.

In Section 4.2 we analyze a single parallelogram of the grid. Given adeformation u with gradient ∇u, we define a finite number of values, corre-sponding to averages over cells and edges, which define a projection of u ontoa discrete space as in a finite-element analysis. The gradient structure of ∇utranslates into compatibility conditions for the averages. If the deformationu has vanishing energy, these compatibility conditions imply that the pro-jection of u coincides with the projection of an affine function. The discretecharacter of the estimate allows a perturbation argument and we concludethat, for small-energy deformations u, the projection is close to the one ofan affine function.

In Section 4.3 we combine those results to construct, by means of piece-wise bilinear interpolation, a new continuous deformation, whose energy iscontrolled by the original one, and which uses only one well. Since the gridrefines towards the chosen line, the two maps coincide on the line. ThusKorn’s inequality yields the H1/2 bound.

The final result of this section is the following rigidity estimate.

Proposition 4.1. Given Ω = (−d, d) × (−l, l) there are constants η0, c > 0depending on W , M , l and d, such that, for every ε > 0 and every u : Ω =(−d, d) × (−l, l) → R

2 such that, for some J ∈ A,B,

Iε[u,Ω] ≤ η ≤ η0 , and ‖∇usym − J‖L2(Ω) ≤ η ,

there is a subset Y ⊂ [−l, l] of measure L1(Y) ≥ l with the following property.

14

(a) (b)

Figure 4.1: (a): Sketch of the simplified self-similar grid. (b): inclusion ofdiagonals and subdivision of each stage into three.

For every y0 ∈ Y there is an affine function wy0 : R2 → R

2 with ∇wsymy0

=J , such that on the slice Σy0 = (−d/2, d/2) × y0

‖u(·, y0) − wy0(·, y0)‖2H1/2(Σy0 ) ≤ cεη.

By an affine change of variables we can assume that J = A, that A and Bhave the form given in (2.6), and Ω = (−1, 1)2. By standard approximationarguments it suffices to consider functions u of class C2.

Our argument is basically conceived on a square grid which refines geo-metrically towards the chosen value of y, as shown in Figure 4.1(a), and suchthat on all grid edges only one of the wells is used. In order to obtain enoughlinear equations to uniquely determine the averages on all grid segments, weneed however to consider also averages on diagonals, and to subdivide eachrefinement step into n parts (n ≥ 4), see Figure 4.1(b).

In choosing the grid we exploit the smallness of the set ΩB where ∇uis in the B-phase, and the smallness of its perimeter. In general, it is notpossible to find a translation of a rigid grid as in Figure 4.1(b), such thatthe edges do not intersect ΩB. This follows from a result of Komjath [21]:For every d ∈ (0, 1) and δ > 0, there is a countable union of intervals E =∪i(ai − ri, ai + ri) ⊂ (0, 1), with |E| < δ, such that for every y ∈ (0, 1),there is a k ∈ N with y − d−k ∈ E. Applied to our situation, let E ′ =∪iB((0, ai), ri) ⊂ (−1, 1) × (0, 1). The set E ′ has perimeter and area lessthan πδ. Nevertheless, every translation of a rigid grid as in Figure 4.1(b)hits E ′. We shall therefore generalize to grids which are slightly tilted andwhose y-spacing fluctuates by a small amount, as in Figure 4.2 (a). This willgive two degrees of freedom (tilt and spacing) at each level.

We next give the precise definition of the grid. A one-grid on the unit

15

Figure 4.2: (a) Global structure of the grid constructed in Lemma 4.3.(b) One step in the grid, for n = 3. The grid is composed by the union ofthe sides and the diagonals of 3 layers of 3 · 2k equal parallelograms. Thehorizontal side of each of them is dk = 2−k, the height (yk − yk−1)/3. Thelower-left corner of the whole grid is (xk, yk), the upper left one (xk+1, yk1

).

square consists of the union of the four sides and the two diagonals,

G(1) =

(x, y) ∈ [0, 1]2 : (x, y) ∈ ∂[0, 1]2 or x = y or x = 1 − y

, (4.1)

see Figure 4.3(a). A n-grid on the unit square consists of the union of n2

such grids scaled and translated to the n2 subsquares of side 1/n,

G(n) =n−1⋃

i,j=0

1

nG(1) +

(

i

n,j

n

)

,

see Figure 4.3(b). A n-grid on a parallelogram is the image of the n-grid ona square under the affine transformation T that maps the square onto theparallelogram, see Figure 4.3(c). We shall only use parallelograms with oneside parallel to the x axis, and parametrize them with the positions z and z ′

of the leftmost points of the horizontal sides, which at stage k have lengthdk = 2−k. Let Tk(z, z

′) be the affine map that brings the unit square onsuch a parallelogram. Then, the k-th stage of the grid, as sketched in Figure4.2(b), is given by

G(n)k (z, z′) = Tk(z, z

′)2k−1⋃

i=0

[

G(n) +

(

i0

)]

. (4.2)

Given a sequence of points zk, we construct the grid G as the union of thesubgrids G

(n)k (zk, zk+1) over k ∈ N. The discussion of the following sections

is formulated for a generic n larger than or equal to 4, but the index n issuppressed in many expressions. The result is used only for the case n = 4.

16

(a) (b) (c)

Figure 4.3: Grids. (a): a 1-grid on a square. (b): a 3-grid on the samesquare. (c): a 3-grid on a parallelogram.

4.1 Construction of a small-energy grid

We perform the construction for horizontal cross-sections in the upper halfof the domain, i.e. y > 0, the other case can be obtained by reflection. Thefirst step is to derive an L∞ bound for the one-dimensional Modica-Mortolaproblem. Let f(x, y) = 1

εW (∇u) + ε|∇2u|2 be the energy density.

Lemma 4.2. For any small σ > 0 there is d = d(σ) (implicitly dependingalso on W and M), such that the following holds:

(i). If γ is a piecewise C1 curve such that∫

γf dH1 ≤ d, and there is z0 ∈ γ

such that |∇usym(z0) − A| ≤ σ/2, then |∇usym − A| ≤ σ on the entiresegment γ. The same holds for B.

(ii). If γ is a piecewise C1 curve such that∫

γf dH1 ≤ d and length |γ| ≥ ε,

then either |∇usym − A| ≤ σ on the entire curve, or |∇usym − B| ≤ σon the entire curve.

Proof. The first statement is essentially the Modica-Mortola compactnessresult. Indeed, let γ(s) be a parametrization of γ, and dW as in (2.10). Then

∫

γ

f ≥∫ 1

0

√

W (∇u(γ(s)))∣

∣

∣

∣

d

ds∇u(γ(s))

∣

∣

∣

∣

ds ≥ dW (∇u(z),∇u(z′))

for all z, z′ ∈ γ. We set

dA = inf dW (F,G) : |F sym − A| ≤ σ/2, |Gsym − A| ≥ σ

and analogously dB, which are positive since W is nonzero away from thewells. Hence part (i) holds for any d ≤ 1

2min(dA, dB).

17

To obtain the second statement, let

d∗ =1

2inf W (F ) : |F sym − A| ≥ σ/2, |F sym −B| ≥ σ/2 ,

and set d = min(dA, dB, d∗). It is clear that there must be a point on γ whereW (∇u) ≤ d∗. Then we conclude using part (i).

Lemma 4.3. Given θ ∈ (0, 1) and δ ∈ (0, 1/4), there are constants η0, ε0, c >0, such that for every ε ∈ (0, ε0) and u ∈ C2((−1, 1)2,R2) with

Iε[u, (−1, 1)2] ≤ η ≤ η0 , and ‖∇usym − A‖2L2((−1,1)2) ≤ η ≤ η0, (4.3)

there is a subset Y ⊂ (0, 1) of measure L1(Y) ≥ θ of ’good’ positions y.This means that for each y0 ∈ Y we can find a sequence of points zk =

(xk, yk) such that, with dk = 2−k, there holds for all k

(i). yk ∈ [y0 − dk, y0 − dk + δdk], |xk − xk+1| ≤ δ2−k, −1 < xk < −1 + 3δ,

(ii). Iε[u, (−1, 1) × (yk, y0)] ≤ cη|yk − y0|,

(iii). On each point of G(n)k (zk, zk+1), we have |∇usym − A| ≤ δ,

(iv). The line energy satisfies

∫

G(n)k (zk,zk+1)

|∇usym − A|2dH1 ≤ cηε ,

(v). The rectangle (−1/2, 1/2) × (y0 − 1/2, y0) is contained in the union ofthe convex envelopes of the Gk’s.

Proof. The idea of the proof is the following. The integral of the energydensity f on most segments is small, due to the first inequality of (4.3).Therefore, by Lemma 4.2 on each such segment only one phase can be used,and if they form a connected set it is the same everywhere. The secondinequality of (4.3) then implies that this phase is A. The main difficulty,which renders the proof rather technical, is that we need to choose an infinitenumber of segments, which satisfy simultaneously a number of properties.

In the proof we often have to show that if a finite list of estimates issatisfied on average, then - up to a constant - there are many points whereall of them are satisfied. More precisely, if ψi : (0, 1) → R, 1 ≤ i ≤ N arefinitely many nonnegative functions which satisfy

∫

ψi ≤ ci, and θ is anynumber in (0, 1), then there is E ⊂ (0, 1), with |E| ≥ θ such that for allx ∈ E and all i we have ψi(x) ≤ Nci/(1 − θ). Indeed, if this were not the

18

case, then the set (0, 1)\E, on which the nonnegative function ψ =∑

i ψi/ciis larger than N/(1− θ), would be larger than 1− θ, hence the integral of ψon (0, 1) would be larger than N . On the other hand the assumption on theψi gives immediately

∫

(0,1)

∑

i ψi/ci ≤ N , a contradiction.

Step 1. Choice of x. We first show that there is a vertical line, x ×(−1, 1), such that

|∇usym(x, y) − A| ≤ δ/4 for all y . (4.4)

Inequality (4.4) follows from Lemma 4.2(ii), provided that we choose xso that, with κ1 = d(δ/4) from Lemma 4.2 and κ2 = κ2(W ), there holds

∫ 1

−1

f(x, y) dy ≤ κ1, and

∫ 1

−1

|∇u(x, y) − A|2 dy ≤ κ2.

Since the integrals over x of both quantities are controlled by (4.3), by choos-ing η0 sufficiently small, there must be some x for which this is true. We notethat η0, here and in the following, depends only on δ and W .

Step 2. Choice of y0. We claim that for any θ1 < 1, there is a θ1-fractionof points y ∈ (0, 1) such that, for some c1 = c1(θ1), the following holds:

for all k ∈ N ,1

dk

∫ y

y−dk

(∫ 1

−1

f(x, y′) dx

)

dy′ ≤ c1η , (4.5)

To see this, consider the intervals (y − dk, y) where (4.5) does not hold,and scale them up by (1 + 2ξ), where ξ is a small positive quantity. LetF = (yi − dki

(1 + ξ), yi + ξdki) ∩ (0, 1) be the resulting family of intervals.

The family of intervals F covers some subset I of (0, 1), which contains ally for which (4.5) does not hold. By Vitali’s covering Lemma there is asubfamily G ⊂ F of disjoint intervals covering at least one fifth of I. Then

η ≥∫

(−1,1)×I

f ≥∑

Ij∈G

∫

(−1,1)×Ij

f ≥∑

Ij∈G

|Ij|1 + 2ξ

c1η ≥ |I|c15(1 + 2ξ)

η ,

which gives |I| ≤ 5(1+2ξ)/c1. We fix ξ = 1/2 and c1 such that 1−θ1 < 10/c1.Then |I| ≤ 1 − θ1, and the claim follows.

Since f is continuous we further obtain∫

f(·, y0) ≤ c1η. If we choose η0

small compared to d(δ/2) (as defined in Lemma 4.2) we further have that

|∇usym(x, y0) − A| ≤ δ/2 for all x ,

since we can use (4.4) in the point (x, y0).

19

Step 3. Choice of the horizontal lines. We seek yk close to y0 − dk (inthe sense of point (i) of the statement) such that

∫ 1

−1

f (x, yk) dx ≤ c2η . (4.6)

Further, if k > 1 we also require the same bound for n− 1 intermediate linesbetween yk and yk−1, namely,

∫ 1

−1

f

(

x, yk −i

n(yk − yk−1)

)

dx ≤ c2η (4.7)

for i ∈ 1, 2, . . . , n − 1. This is always possible, since by (4.5) the integralof the quantity in (4.6) over the set of admissible yk, which has width δdk,is controlled by c1ηdk. Hence we can choose one yk so that (4.6) holds withc2 = c1/δ. The same argument applied to the sum of the n integrals appearingabove permits to choose the final yk, which satisfies (4.6) and (4.7) with alarger value of c2. It is crucial that, in selecting yk, we have to satisfy only afinite number of conditions.

Step 4. Choice of the non-horizontal lines. We denote by Gk(x, y, x′, y′)

the union of all segments of one refinement level, as defined in (4.2), and byGnhk (x, y, x′, y′) the union of all non-horizontal ones, which are the only ones

of interest here (the horizontal ones have already been treated in Step 3).We remark that for all arguments satisfying condition (i) in the proposition,a grid Gnh

k is the union of c2k segments, whose total length is uniformlybounded, and which form an angle larger than π/8 with the horizontal axis.We shall choose x1 ∈ I1 = (−1 + δ,−1 + 2δ) and, at stage k ≥ 1, xk+1 ∈Ik+1 = xk + (−dkδ, dkδ) such that

∫

Gnhk (xk,yk,xk+1,yk+1)

fdH1 ≤ c3η . (4.8)

The key observation is that, for all k, and all tk ∈ (−δdk, δdk), we have

∫

Ik

∫

Gnhk (xk,yk,xk+tk,yk+1)

fdH1 dxk ≤ c

∫

(−1,1)×(yk,yk+1)

f ≤ c1dkη .

Hence by choosing c3 large compared to c1, for a given tk we find a largeset of xk ∈ Ik such that (xk, xk + tk) defines a low-energy strip in the senseof (4.8) (here and below ’large set’ means a set of measure at least 8/9|Ik|).Moreover, we will see that for a large set of xk there is a large set of tk suchthat the couple (xk, xk + tk) defines a low-energy strip, and iterate.

20

We now make the inductive argument rigorous. Each xk must be in aninterval Ik, with |Ik| = 4δdk. We define the set of ’good’ values of xk as

Xk = xk ∈ Ik : (4.8) holds for at least two-thirds

of the xk+1 = xk + tk, tk ∈ (−δdk, δdk) ,

and show that Xk covers at least two-thirds of the interval Ik. To see this, letP be the set of pairs (xk, tk), seen as a subset of J = Ik×(−dkδ, dkδ), for which(4.8) holds. By the argument above each horizontal section x : (x, tk) ∈ Pof P covers eight-nineth of the corresponding section of J , therefore the areaof P is at least eight-nineth that of J . Then also two-thirds of the verticalsections of P have one-dimensional volume at least two-thirds. If not, thetotal volume were |P | < (2/3) · 1 + (1/3) · (2/3) = 8/9, a contradiction.

At the first step, we choose freely one x1 in the large set X1. In theinduction step we are given xk ∈ Xk and want to choose xk+1. Since xk ∈ Xk,for two-thirds of the choices of tk ∈ (−δdk, δdk) the k-th grid, with xk+1 =xk+ tk, satisfies (4.8). On the other hand, since |Xk+1| ≥ (2/3) ·2δdk+1, xk+1

can be chosen to satisfy additionally xk+1 ∈ Xk+1. By induction we find thesequence xkk.

4.2 The linear algebra lemma

We consider a single element of the grid, as defined in (4.1-4.2), and show(Lemma 4.4) that line averages of u on grid edges are approximately affine.In the simplest case this is a unit square, subdivided into n× n subsquares,and the grid is the union of all their sides and diagonals. In general, the gridis defined on a parallelogram P which, modulo a translation and a scaling,has vertices in (0, 0), (1/l, 0), (s, l), and (s + 1/l, l). We first transform Pback to a square. The linear map

T =

(

1/l s0 l

)

(4.9)

maps the unit square onto P . We shall therefore replace u : P → R2 by

u : (0, 1)2 → R2, according to u(x) = T Tu(Tx). The energy of u controls

the squared distance of ∇u from the set Rϕ : ϕ ∈ R ∪ B + Rϕ : ϕ ∈ R,where the infinitesimal rotation Rϕ was defined in (1.5). For the transformedsolution, we have a control of the distance of ∇u from the set

T TRϕT : ϕ ∈ R ∪ T T (B +Rϕ)T : ϕ ∈ R= Rψ : ψ ∈ R ∪ B2sl +Rψ : ψ ∈ R,

21

with Bt = T TBT = ex⊗ ey + ey⊗ ex+ tey⊗ ey, and t = 2sl. By constructionof the grid the parameters t and l can be chosen uniformly close to 0 and1, respectively, hence the original and transformed distances are uniformlyequivalent.

Lemma 4.4. There exist constants σ0, t0, c > 0 such that for every d > 0,on the domain Ω = (0, d) × (0, d), the following holds. Let u ∈ W 2,2(Ω,R2)have small energy and use only phase A on the edges γi, in the sense that

1

d2

∫

Ω

min(

|∇usym|2, |∇usym −Bt|2)

≤ σ ≤ σ0, (4.10)

∑

γi∈dG(n)

1

d

∫

γi

|∇usym|2dH1 ≤ σ ≤ σ0, (4.11)

for some t ∈ [−t0, t0]. Here Bt = ex⊗ ey + ey ⊗ ex + tey ⊗ ey, and we assumen ≥ 4. Then the averages over top and bottom edges,

u+i =

n

d

∫ i+1nd

ind

u(x, d) dx , u−i =n

d

∫ i+1nd

ind

u(x, 0) dx

for 0 ≤ i < n are approximately affine, i.e. there are φ ∈ R and w0 ∈ R2

such thatn−1∑

i=0

∣

∣u+i − w+

i

∣

∣

2+∣

∣u−i − w−i

∣

∣

2 ≤ cσd2 , (4.12)

where

w+i = w0 +Rφ

(

i d/nd

)

, w−i = w0 +Rφ

(

i d/n0

)

.

Remark 4.5. The statement of Lemma 4.4 remains valid for paralellogramsT (0, d)2, where T was defined in (4.9), with |s| + |l − 1| ≤ t0. Equations(4.10), (4.11) and (4.12) are unchanged, the definition of u±

i now reads

u+i =

n

d/l

∫ sd+ i+1nd/l

sd+ ind/l

u(x, dl)dx , u−i =n

d/l

∫ i+1nd/l

ind/l

u(x, 0)dx

and that of w±i

w+i = w0 +Rφ

(

sd+ i d/nldl

)

, w−i = w0 +Rφ

(

i d/nl0

)

.

22

Proof of Remark 4.5. All areas and lengths are uniformly close to d2 and d.The result follows via application of Lemma 4.4 to the function

u(x) = T Tu(Tx) ,

where T was defined in (4.9).

Proof of Lemma 4.4. After scaling we can assume d = n so that the gridis composed of unit squares. The general structure of the argument is thefollowing: we first define a finite number (depending only on n) of discretevariables, which are averages of ϕ and of χΩB

over triangles of the grid. Thenwe derive a finite number of linear compatibility conditions that must besatisfied. This gives a system of linear equalities and inequalities. In the rigidcase σ = 0 we show that there is no nontrivial solution to this system, i.e.,the averages must coincide with the averages of an affine function. Finally,since the finite-dimensional system of equalities and inequalities allows for asmall perturbation, we conclude for positive σ the quantitative result.

We denote by ΩB the subset of Ω where |∇usym − Bt| < |∇usym| minusthe edges of the grid, and by χB its characteristic function. Further, let ϕ(z)be the angle associated with ∇u(z), i.e. 2ϕ = (∇u)12−(∇u)21. Assumptions(4.10) and (4.11) state that

∇u−Rϕ −BtχB = O(√σ) (4.13)

in the L2-sense over volumes and in the L2-sense over edges. By the analysisof the finite dimensional system in the rest of this subsection, this implies(see (4.29)) that there is φ ∈ R such that all integrals of ϕ(x)− φ and of χBover subsquares Qij are of order

√σ. Then, using again (4.13), we get

∫

Qij

(∇u−Rφ) = O(√σ), (4.14)

and therefore the lemma.

4.2.1 The set of linear equations

In order to derive the linear equations for averages we study the case σ = 0.The square Ω = (0, n)2 ⊂ R

2 consists of squares Qij = (i, i + 1) × (j, j +1), i, j = 0, ..., n−1. The geometry is chosen such that on edges of squares andon diagonals D+

ij = (i, j) + (t, t)|t ∈ (0, 1) and D−ij = (i, j) + (t, 1 − t)|t ∈

(0, 1) the gradient lies in the A-well, i.e. ∇u = Rϕ.

23

Figure 4.4: Labeling of vertices and of subtriangles in each square.

Variables. In every square Qij, for 0 ≤ i, j < n, we define four triangles ofthe form (i+1/2+x, j+1/2+y)|(x, y) ∈ (−1/2, 1/2)2, (x, y)·(±ex±ey) > 0.We order them such that triangle 1 is in the upper right and triangle 4 is inthe lower right (see Figure 4.4). In every triangle Qijk we define

ϕijk =

∫

Qijk

ϕ, bijk =

∫

Qijk

χB.

For brevity we do not separate the three indices of ϕ and b. For example,ϕi+1j+13 refers to the 3-triangle of square (i+1, j+1). We use these 8 variablesper square for our calculations. In the squares along the left boundary of Ωwe use only the triangles labelled 1 and 4, in squares on the right boundarywe use only the triangles labelled 2 and 3. For σ = 0,

∇u(z) = Rϕ(z) +BtχB(z) =

(

0 −ϕ+ χBϕ+ χB tχB

)

.

Mass equations. By definition, in interior squares

ϕij1 + ϕij3 = ϕij2 + ϕij4, (4.15)

bij1 + bij3 = bij2 + bij4. (4.16)

Volume averages. We express the difference of averages of u in two ways.

IV :=

∫ 1

0

u(x, 1) dx−∫ 1

0

u(1, y) dy =

∫

Q1

∇u ·(

−11

)

=

∫

Q1

[

Rϕ(z) +BtχB(z)]

·(

−11

)

dz =

(

−ϕ1 + b1−ϕ1 − (1 − t)b1

)

.

We can calculate the same term in a different way.

IV =

∫

Q2

∇u ·(

01

)

+

∫

Q4

∇u ·(

−10

)

=

(

−ϕ2 + b2tb2 − (ϕ4 + b4)

)

.

24

We find, in interior squares,

ϕ1 − b1 = ϕ2 − b2 (4.17)

ϕ1 + (1 − t)b1 = ϕ4 + b4 − tb2 . (4.18)

Therefore averages of ϕ − b are the same for indices 1 and 2. By the massequations, the same holds for indices 3 and 4.

Line averages on horizontal lines. We next compare volume averageswith line averages. For line averages we use the hat function ψ : (−1, 1) → R,ψ(x) = 1 − |x| with integral 1. For 0 ≤ i < n− 1 we get

IL =

∫ i+2

i+1

u(x, j + 1) dx−∫ i+1

i

u(x, j + 1) dx

=

∫ i+2

i

ψ(x− (i+ 1))∇u(x, j + 1) ·(

10

)

dx =

(

0ϕ

)

,

where ϕ is the weighted average of ϕ over the horizontal line of length 2. Wecan calculate the same difference IL with the help of two volume integrals.

IL =

∫

Qij1

∇u ·(

1−1

)

+

∫

Qi+1j2

∇u ·(

11

)

=

(

ϕij1 − bij1ϕij1 + (1 − t)bij1

)

+

(

−(ϕi+1j2 − bi+1j2)ϕi+1j2 + (1 + t)bi+1j2

)

.

Comparing the expressions for the first component of IL yields

(ϕ− b)ij1 = (ϕ− b)i+1j2. (4.19)

An analogous calculation using volume integrals over triangles lying abovethe line gives

(ϕ− b)ij4 = (ϕ− b)i+1j3. (4.20)

Together with (4.17) this shows that the quantity ϕ− b is horizontally con-stant across the entire grid.

Line averages on vertical lines. We set

IvL =

∫ j+2

j+1

u(i, y) dy −∫ j+1

j

u(i, y) dy

=

∫ j+2

j

ψ(y − (j + 1))∇u(i, y) ·(

01

)

dy =

(

−ϕ0

)

,

25

where ϕ is a weighted average of ϕ over the vertical line of length 2. We canevaluate it with averages over volumes lying to the right of the vertical line.

IvL =

∫

Qij2

∇u ·(

11

)

+

∫

Qij+13

∇u ·(

−11

)

=

(

−ϕij2 + bij2ϕij2 + (1 + t)bij2

)

+

(

−ϕij+13 + bij+13

−ϕij+13 − (1 − t)bij+13)

)

.

The first representation implies that the second component vanishes,

ϕij+13 + (1 − t)bij+13 = ϕij2 + (1 + t)bij2. (4.21)

In the same way one shows

ϕij1 + (1 − t)bij1 = ϕij+14 + (1 + t)bij+14. (4.22)

We conclude that for t = 0 the quantity ϕ+ b is vertically constant.

Line averages on upward diagonals. We calculate along the diagonal

I+D :=

∫

D+i+1j+1

u−∫

D+ij

u =

∫ 2

0

ψ(x− 1)∇u(i+ x, j + x) ·(

11

)

dx

=

∫ 2

0

ψ(x− 1)

(

−ϕ(i+ x, j + x)ϕ(i+ x, j + x)

)

dx =

(

−ϕϕ

)

,

and we evaluate the same expression with volume averages.

I+D :=

∫

Qij4

∇u ·(

10

)

+

∫

Qi+1j3

∇u ·(

10

)

+

∫

Qi+1j1

∇u ·(

01

)

+

∫

Qi+1j+14

∇u ·(

01

)

=

(

(b− ϕ)i+1j1 + (b− ϕ)i+1j+14

(ϕ+ b)ij4 + (ϕ+ b)i+1j3 + tbi+1j1 + tbi+1j+14

)

.

The first representation shows that the two components add to zero, hence

ϕi+1j1 − (1 + t)bi+1j1 + ϕi+1j+14 − (1 + t)bi+1j+14

= (ϕ+ b)ij4 + (ϕ+ b)i+1j3.(4.23)

Using instead integrals over volumes above the diagonal leads to

ϕij2 − (1 + t)bij2 + ϕij+13 − (1 + t)bij+13

= (ϕ+ b)ij+11 + (ϕ+ b)i+1j+12.(4.24)

26

Line averages on downward diagonals. An analogous calculation gives

ϕij2 − (1 − t)bij2 + ϕij+13 − (1 − t)bij+13 = (ϕ+ b)i+1j3 + (ϕ+ b)ij4 (4.25)

and

ϕi+1j1 − (1 − t)bi+1j1 + ϕi+1j+14 − (1 − t)bi+1j+14

= (ϕ+ b)ij+11 + (ϕ+ b)i+1j+12.(4.26)

4.2.2 The abstract form of the equations.

We can write equations (4.15–4.26) and the condition χB ≥ 0 in the form

Mt ·(

ϕb

)

= 0, b ≥ 0,

where ϕ and b are the vectors with components ϕijk and bijk, respectively.

The case t = 0. Equation (4.23) with t = 0 simplifies to

(ϕ− b)i+1j1 + (ϕ− b)i+1j+14 = (ϕ+ b)ij4 + (ϕ+ b)i+1j3. (4.27)

The left hand side is independent of i since ϕ − b is horizontally constantin the grid. Moreover, the right hand side is independent of j, since byequations (4.18), (4.21), and (4.22) the quantity ϕ+ b is vertically constant.We conclude that both sides in equality (4.27) are constant in all interiorcells. After a normalization of ϕ we can assume that both sides in (4.27)vanish on all interior squares.

We add up over four triangles with a common vertex,

0 = (ϕ− b)ij1 + (ϕ− b)ij+14 + (ϕ− b)i+1j2 + (ϕ− b)i+1j+13

= (ϕ+ b)ij1 + (ϕ+ b)i+1j2 + (ϕ+ b)ij+14 + (ϕ+ b)i+1j+13

− 2[bij1 + bi+1j2 + bij+14 + bi+1j+13]

= −2[bij1 + bi+1j2 + bij+14 + bi+1j+13].

In this calculation we used once more that ϕ− b is horizontally constant andthat ϕ+ b is vertically constant.

Since all b are non-negative, necessarily b vanishes in all interior squares.Then with (ϕ− b) also ϕ is horizontally constant and with (ϕ+ b) ϕ is alsovertically constant. Therefore ϕ vanishes identically in the interior of thegrid. With the help of the diagonal equalities we conclude that b and ϕvanish also in the triangles in boundary squares that meet an interior cell atan edge. This shows that for t = 0 the system

M0 ·(

ϕb

)

= 0, b ≥ 0

has only the trivial solution (ϕ, b) = 0.

27

The case t 6= 0. For t = 0 the discrete system has only the trivial solution.We claim for the case t 6= 0 with |t| small:

(i). The system

Mt ·(

ϕb

)

= 0, b ≥ 0

has only the trivial solution (ϕ, b) = 0.

(ii). With c independent of t every solution of

Mt ·(

ϕb

)

= f, b ≥ 0 (4.28)

satisfies an estimate‖(ϕ, b)‖ ≤ c‖f‖. (4.29)

Both claims follow immediately by contradiction. (i). Assume that fora sequence tn → 0 there are solutions xn = (ϕn, bn) of Mtnxn = 0 with‖xn‖ = 1 and Bxn := bn ≥ 0. Since xn has finitely many components,we can choose a convergent subsequence and conclude for the limit x0 thatM0x0 = 0, Bx0 ≥ 0. A contradiction to ‖x0‖ = 1.

(ii). We again assume the contrary. Then for a sequence tn → t0 thereare fn → 0 and solutions xn of Mtnxn = fn with ‖xn‖ = 1 and Bxn ≥ 0. Wechoose a convergent subsequence and conclude for the limit x0 that Mt0x0 =0, Bx0 ≥ 0. Because of ‖x0‖ = 1 this contradicts the result of (i).

Conclusion. It remains to show that f in the finite-dimensional equation(4.28) is of order

√σ. This follows from the fact that equations (4.15)–

(4.26) hold up to error terms f that consists of volume integrals and lineintegrals of F = ∇u − Rϕ − BtχB, which are of order

√σ by (4.13). We

show this concretely for (4.19), the other ones are analogous. Again, the twoexpressions for the difference of line-averages of u must coincide,

0 =

∫ i+2

i

ψ(x− (i+ 1))∇u(x, j + 1) ·(

10

)

dx

−∫

Qij1

∇u ·(

1−1

)

−∫

Qi+1j2

∇u ·(

11

)

.

Inserting ∇u = Rϕ +BtχB + F , the first component gives

0 = −(ϕ− b)ij1 + (ϕ− b)i+1j2 +

∫ i+2

i

ψ(x− (i+ 1))F11

−∫

Qij1

(F11 − F12) −∫

Qi+1j2

(F11 + F12).

28

Relation (4.13) for integrals of F yields f = O(√σ) in the finite-dimensional

system (4.28). Therefore inequality (4.29) proves (4.14) and the lemma.

4.3 Proof of the H1/2 bound

So far we have achieved the following. Given a function u with low energywe found a grid such that (i) ∇u is in the A-phase along the edges of thegrid, and (ii) the energy is small at all refinement stages. The linear algebralemma assures that averages of the function u are approximately affine inevery cell of the grid. Let uΣ be the restriction of u to the line y = y0. Weshall construct an extension of u|Σ across the entire grid which is H1-closeto an affine function, which will imply the H1/2 bound on uΣ.

We use the grid constructed in Section 4.1, for n = 4, and assume y0 = 0.The corners of the grid have the coordinates (xk,i, yk), with i = 0, ..., 2k−1,k ≥ 1, and xk,i = xk + idk. The grid parallelogram with vertices (xk,i, yk),(xk,i + dk, yk), (xk+1,2i, yk+1) and (xk+1,2i + dk, yk+1) is denoted by Qk,i, it hasthe height lk = |yk+1 − yk| and the width dk = 2−k. Note that lk and dk arealways comparable in size. Let

uk,i =1

|γk,i,1|

∫

γk,i,1

u

be the average of u over the first part of the bottom edge of cell Qk,i, γk,i,1 =(xk,i, xk,i + dk/n) × yk, (uk,i was called u−1 in the previous section), and

E(u,Ω) =

∫

Ω

min(

|∇usym|2, |∇usym −B|2)

.

The linear algebra lemma (Lemma 4.4 and Remark 4.5) implies for verticalfinite differences that with an appropriate angle φk,i

∣

∣

∣

∣

uk+1,2i − uk,ilk

−Rφk,i

(

(xk+1 − xk)/lk1

)∣

∣

∣

∣

2

≤ c

d2k

E(u,Qk,i) +c

dk

∑

j

∫

γk,i,j

|∇usym|2.

In order to derive the analogous estimate for horizontal finite differenceswe first have to study variations of φk,i across the grid. To this end we applyLemma 4.4 three times; in two neighboring macrocells Qk,i and Qk,i+1, andin a collection of n × n cells that form a macrocell and overlaps the othertwo. Since u is approximately affine in all the macrocells we find for angle

29

differences the same estimate as for finite differences,

|φk,i − φk,i+1|2 ≤ ci+1∑

i′=i

(

1

d2k

E(u,Qk,i′) +1

dk

∑

j

∫

γk,i′,j

|∇usym|2)

.

In particular, we find for horizontal finite differences

∣

∣

∣

∣

uk,i+1 − uk,idk

−Rφk,i

(

10

)∣

∣

∣

∣

2

≤ ci+1∑

i′=i

(

E(u,Qk,i′)

d2k

+1

dk

∑

j

∫

γk,i′,j

|∇usym|2)

.

Since the grid refines for increasing k, we also have to compare the valueuk+1,2i+1 with its counterpart on y = yk. We use that φk,i and φk,i+1 areclose to conclude that the averages uk,i,j are approximately affine also acrosstwo neighboring cells Qk,i and Qk,i+1. This yields

∣

∣

∣

∣

uk+1,2i+1 − 12(uk,i + uk,i+1)

lk−Rφk,i

(

(xk+1 − xk)/lk1

)∣

∣

∣

∣

2

≤ c

d2k

E(u,Qk,i) +c

dk

∑

j

∫

γk,i,j

|∇usym|2.

We can now define an interpolation u as follows. In the vertices (xk,i, yk)we set u(xk,i, yk) = uk,i. We then define u on the line Σk := (x, y) : y = ykas the linear interpolation of these point-values. In the strip between Σk andΣk+1 we consider segments with end-points (xk + t, yk) and (xk+1 + t, yk+1),and define u on each such segment as the linear interpolation between itsvalues in the end-points. This results in a bilinear interpolation, with respectto deformed coordinates, in each half-cell T±

k,i. We have here subdivided eachcell Qk,i into two parallelograms with common side joining (xk,i + dk/2, yk)and (xk+1,2i+1, yk+1), and called them T−

k,i and T+k,i.

Lemma 4.6. Let u be as in Proposition 4.1, the grid as above, and let Ω′ bethe union of all the Qk,i. Then, there are u ∈ R

2 and φ ∈ R such that theinterpolation u satisfies the estimate

‖u(z) − u−Rφz‖2H1(Ω′) ≤ cεη .

Proof. In the single cell T±k,i the function u is the bilinear interpolation of

four values which are approximately affine with gradient Rφk,i. Therefore,

with the approximate angle

ϕ(z) =∑

k,i

∑

α∈+,−

φk,iχTαk,i

(z),

30

we find the estimate

‖∇u−Rϕ‖2L2(Ω) =

∑

k,i,±

∫

T±

k,i

|∇u−Rφk,i|2

≤ c∑

k,i

d2k

(

∣

∣

∣

∣

uk+1,2i − uk,ilk

−Rφk,i

(

(xk+1 − xk)/lk1

)∣

∣

∣

∣

2

+

∣

∣

∣

∣

uk+1,2i+1 − 12(uk,i + uk,i+1)

lk−Rφk,i

(

(xk+1 − xk)/lk1

)∣

∣

∣

∣

2

+

∣

∣

∣

∣

uk,i+1 − uk,idk

−Rφk,i

(

10

)∣

∣

∣

∣

2)

≤ c∑

k,i

d2k

(

1

d2k

E(u,Qk,i) +1

dk

∑

j

∫

γk,i,j

|∇usym|2)

= c∑

k,i

E(u,Qk,i) + c∑

k,i,j

dk

∫

γk,i,j

|∇usym|2

≤ cE(u,Ω) + c∑

k

dkεη ≤ cεη.

In the last line we used Lemma 4.3(iv) and the definition of Iε.We have shown that ∇u is L2-close to the infinitesimal rotation Rϕ, in

particular,‖∇usym‖2

L2(Ω) ≤ cεη.

An application of Korn’s inequality yields the desired estimate for u ∈ H 1.

Proof of Proposition 4.1. We transform the domain to the square (−1, 1)2

with a diagonal change of variables and a scaling of u to ensure that theform of B is unchanged, and perform the corresponding redefinition of W .We only do the proof for positive y0, the other case is symmetric. By Lemma4.3 we can construct, for most y0, a grid (with n = 4) entirely contained inΩ and such that it covers the set Ω = (−1/2, 1/2) × (−1/2, y0). By Lemma4.6 we find that the distance of u from an affine function on Ω is controlledby its energy. An application of the trace theorem yields the result.

5 Recovery sequence

In this section we show that for any function u0 on which the limiting func-tional I0 is finite, and for any sequence εi → 0, we can find a sequence

31

uεi→ u0 in L1 such that Iεi

[uεi] converges, as εi → 0, to I0[u0]. This is done

in several steps. By general arguments, we can reduce to the case that u0

has finitely many interfaces. Then, the construction uses modifications of anoptimal sequence for the lim inf of single-interface problems as considered inSection 3. In order to glue together several interfaces, we need to modifythe profile so that it achieves affine boundary conditions. This is done inProposition 5.2 combining the rigidity result of the previous section with anexplicit construction, which is given in Lemmas 5.3 and 5.4. However, atthis stage we only know that there is one sequence εi such that the construc-tion is feasible. In order to show that one can do the construction for anysequence εi → 0, we provide an additional argument which uses optimal se-quences with affine boundary conditions in rectangles with large aspect ratio(Proposition 5.5).

We start with the main result which we derive with the help of Remark5.6.

Proposition 5.1 (Upper bound). Let Ω be an open, bounded, strictly star-shaped set in R

2, and u ∈ W 1,2(Ω,R2) be such that ∇usym ∈ BV (Ω, A,B).Then, for any sequence εi → 0, there is a sequence ui → u in L1(Ω), suchthat

limi→∞

Iεi[ui,Ω] = I0[u,Ω] ,

with I0 as in (1.6).

Proof. By the change of variables discussed in Section 2, we can assume theinterface normals to be νh = ey and νv = ex, with A and B of the formdiscussed there. This will be tacitly assumed in the rest of this section.

The strategy of the proof is the following. First we exploit star-shapednessto replace u with a scaled version uη, which has interfaces contained in finitelymany disjoint rectangles (see below). Then we use for each εi Remark 5.6inside each of the rectangles, and an affine function outside, to obtain afunction that converges to uη and has comparable energy. The conclusionfollows by taking a diagonal subsequence.

Given η > 1, we consider the rescaling

uη(z) = ηu

(

z

η

)

.

Since the domain is star-shaped, uη(z) is well defined for z ∈ Ω, and I0[uη,Ω] ≤ηI0[u,Ω]. Let Si denote the segments composing the jump set of u. Thejump set of uη is the union of the sets Sηi = ηSi ∩ Ω, each of which is a(possibly empty) union of collinear segments. We now show that for i 6= j,

32

dist(Sηi , Sηj ) > 0 if they are both nonempty. Indeed, by Proposition 2.2, the

closures Si and Sj can intersect only in end-points zij ∈ ∂Ω, therefore, bystrict star-shapedness of Ω, ηSi and ηSj can intersect only outside Ω.

We next show that only finitely many of the Sηi are nonempty. If not,u had an infinite number of interfaces Si with Si ∩ (Ω/η) 6= ∅. We choosepoints zi ∈ Si∩ (Ω/η) and, taking a subsequence, we get zi → z0 ∈ Ω/η ⊂ Ω,the last inclusion holds by strict star-shapedness of Ω. Since the total lengthof the interfaces is bounded, along any infinite sequence the length of theinterfaces Si converges to zero. Therefore the distance dist(zi, ∂Ω) convergesto zero, hence z0 ∈ ∂Ω, a contradiction.

We can construct rectangles ωαi such that each contains one of the Sηi ,and the sides orthogonal to Sηi do not intersect Ω. Here and below we use α ∈h, v to label horizontal and vertical interfaces, and focus on the horizontalones, the vertical ones can be treated analogously. We consider the (finitelymany) segments Si with Sηi = ηSi∩Ω nonempty. Let (x±i , yi) be the endpointsof Si, which belong to ∂Ω. The points (ηx±i , ηyi) are not contained in Ω, andthere are finitely many of them, hence there is σ > 0 such that the segmentsηx±i × (ηyi−σ, ηyi+σ) do not intersect Ω. We choose the latter as verticalsides of the rectangles, which take the form

ωhi = (ηx−i , ηx+i ) × (ηyi − σ, ηyi + σ) .

If σ is small enough, the ωαi are all disjoint.We claim that a sequence uη,i can be found such that

uη,i → uη in L1(Ω) , lim supi→∞

Iεi[uη,i,Ω] ≤ ηI0[u,Ω] . (5.1)

To see this, we consider the set

Ω1 = Ω \⋃

i,α

ωαi .

In each connected component of Ω1 we define ui as uη plus a suitable affinefunction with skew-symmetric gradient; in each of the ωαi we instead use theadaptation of an optimal sequence for the lower bound discussed in Remark5.6, also adding a suitable affine function. The affine connection can alwaysbe chosen appropriately, since Ω is star-shaped. This proves (5.1).

Finally, we choose a sequence ηj → 1, ηj > 1. Since uηj→ u0 in L1 as

j → ∞, taking a diagonal sequence we conclude the proof.

We come to the main construction step, which permits to modify a small-energy sequence for a single-interface problem to obtain one which still hassmall energy and is affine close to the upper and lower boundaries.

33

Proposition 5.2. For any l > 0, d > 0, let εi → 0, ui → u+h be such that

limi→∞

Iεi[ui, (−2d, 2d) × (−l, l)] = 4dkh , lim

i→∞‖ui − u+

h ‖L1 = 0 ,

where u+h was defined in (3.2). Then there is a sequence vi : (−d, d) ×

(−l, l) → R2 such that, for the same εi,

limi→∞

Iεi[vi, (−d, d) × (−l, l)] = 2dkh , lim

i→∞‖vi − u+

h ‖L1 = 0 ,

which satisfies

vi(x, y) =

A(x, y)T if y ≥ l/2

(B + Sh + Si)(x, y)T + ai if y < −l/2

where Sh is as in (3.1), Si is skew-symmetric, Si → 0, ai → 0. The sameholds for vertical interfaces, and for those of the opposite orientation.

Proof. By Lemma 3.2, we get

lim inf Iεi[ui, (−2d, 2d) × (−l/8, l/8)] ≥ 4dkh,

and thereforeIεi

[ui, (−2d, 2d) × (l/8, l)] → 0

(and the same on (−l,−l/8)). Let now

ηi = Iεi[ui, (−2d, 2d) × (l/8, l)] + ‖ui − u+

h ‖2W 1,2((−2d,2d)×(l/8,l)) .

By the compactness result of Theorem 2.1, ηi → 0. By Proposition 4.1applied to (−2d, 2d) × (l/8, l/3), for sufficiently large i, for at least one-halfof the y0 ∈ (l/8, l/3) there are ai, bi, ϕi (depending on y0 and i) such that

∥

∥

∥v

(y0)i

∥

∥

∥

2

H1/2((−d,d),R2)≤ cηiεi ,

where

v(y0)i (x) = ui(x, y0) − A

(

xy0

)

−(

ai − ϕiy0

bi + ϕix

)

. (5.2)

On the other hand, it is easy to see that there is c > 0 such that for at leasttwo-thirds of all y0 in (l/8, l/3) one has

εi‖ui(·, y0)‖2W 2,2((−d,d),R2) +

1

εiIεi

[ui, (−d, d) × (y0, y0 + εi)] ≤ cηi .

34

Then, there is one y0 such that both properties hold. Let

vi(x, y) = ui(x, y) − A

(

xy

)

−(

ai − ϕiybi + ϕix

)

,

where ai, bi and ϕi are as in (5.2). We can apply the construction of Lemma5.4 to the function vi in the domain (−d, d) × (y0, y0 + l/3). The resultingfunction is affine close to the upper boundary of the domain, hence we cancontinue it to (−d, d) × (−l, l). Its energy is controlled by the one of ui plusa constant times ηi. Exactly the same can be done in (−d, d) × (−l,−l/8),after exchanging A with B and a few signs. We have therefore constructeda function wi : (−d, d) × (−l, l), such that

Iεi[wi, (−d, d) × (−l, l)] + ‖wi − uih‖W 1,1 ≤ cηi

and

wi(x, y) =

(A+ Si)(x, y)T + ai if y ≥ 2l/3 ,

(B + Si + Sh)(x, y)T + bi if y ≤ −2l/3 .

Subtracting the affine function Si(x, y)T + ai we get the result.

We now give the explicit construction used in the proof of Proposition 5.2.As in the two-step vertical matching used in [11] for the case of two matrices,we interpolate separately the value of u (Lemma 5.3) and its gradient (Lemma5.4). The treatment of the first interpolation in Fourier space permits to relaxthe assumption of a good H1 control of u on a cross section, which was usedin [11], to an assumption on the H1/2 norm.

Lemma 5.3. Let ε < 1 and u : (−d, d) → R2 be given, with

1

ε‖u‖2

H1/2 + ε ‖u‖2H2 ≤ η . (5.3)

Then for any l > 0 there v : (−d, d) × (0, l) → R2 such that v(x, 0) = u(x),

v(x, y) is constant for y ≥ l/2, and, for c = c(l, d),

1

ε

∫

(−d,d)×(0,l)

|∇v|2 + ε

∫

(−d,d)×(0,l)

|∇2v|2 ≤ cη . (5.4)

Proof. After scaling we can assume l = d = 1. Using a Fourier representationof u, we have

u(x) =∑

k∈πZ/2

ukeikx ,

∑

k

( |k|ε

+ ε|k|4)

|uk|2 ≤ η .

35

Let ψ : (0,∞) → R be smooth, with ψ(0) = 1, ψ(t) = 0 for t ≥ 1, and set

v(x, y) = u0 +∑

k 6=0

ukeikxψ (ky) .

A direct calculation yields (5.4).

Lemma 5.4. Let ε < 1 and u : (−d, d) × (0, l) → R2 be given. Assume that

there is y0 ∈ (0, l/2) such that u(·, y0) : (−d, d) → R2 satisfies (5.3) and

1

ε

∫

(−d,d)×(y0,y0+ε)

|∇u|2 + ε

∫

(−d,d)×(y0,y0+ε)

|∇2u|2 ≤ η .

Then there is w : (−d, d) × (0,∞) → R2 such that w(·, y) = u(·, y) for

y < l/2, w(x, y) constant for y > l, and

1

ε

∫

(−d,d)×(0,∞)

|∇w|2 + ε

∫

(−d,d)×(0,∞)

|∇2w|2 ≤ cη .

Proof. Again, we prove the result by doing an explicit construction for thecase d = l = 1. Let v be the function constructed in the Lemma 5.3, appliedto the set (−1, 1) × (y0, y0 + 1) with u = u(·, y0). Clearly we can continue itas a constant for y > y0 + 1.

Let ψ : R → R be a smooth interpolation function such that ψ(t) = 0 fort ≤ 0, ψ(t) = 1 for t ≥ 1, with |ψ| + |ψ′| + |ψ′′| ≤ c. We define

w(x, y) = v(x, y)ψ

(

y − y0

ε

)

+ u(x, y)

[

1 − ψ

(

y − y0

ε

)]

.

The key ingredient of the estimate is a bound on v − u in L2. To this endwe use the one-dimensional Poincare inequality in the y-direction for fixed xexploiting u = v on y = y0, and then integrate over x to find

∫

(−1,1)×(y0,y0+ε)

(v − u)2 ≤ ε2

∫

(−1,1)×(y0,y0+ε)

|∇v|2 + |∇u|2 .

The first and second derivatives of w can be estimated by

|∇w|2 ≤ c

(

|∇v|2 + |∇u|2 +1

ε2|u− v|2

)

|∇2w|2 ≤ c

(

|∇2v|2 + |∇2u|2 +1

ε2

(

|∇u|2 + |∇v|2)

+1

ε4|u− v|2

)

.

Combining with the previous bound concludes the proof.

36

We now show that the construction can be done for any sequence εi → 0.

Proposition 5.5. For any l > 0, d > 0, and any sequence εi → 0, εi > 0,there are sequences u+

i and u−i such that

limi→∞

Iεi[u±i , (−d, d) × (−l, l)] = 2dkh , lim

i→∞

∥

∥u±i − u±h∥

∥

L1 = 0,

where u±h were defined in (3.2). The same holds for vertical interfaces.

Proof. We give a proof only for horizontal interfaces with positive orientation,the others are clearly equivalent.

Step 1. We claim that there is a sequence vi : (−d, d)×R → R2 such that

lim supi→∞

Iεi[vi, (−d, d) × R] ≤ 2dkh (5.5)

and

for each i, there is Li such that

∇vsymi = A for y > Li

∇vsymi = B for y < −Li .

(5.6)

To prove this, we first observe that by the definition of kh there aresequences εj, uj such that

Iεj[uj, (−2d, 2d) × (−l, l)] → 4dkh , uj → u+

h in L1 .

Let vj be as in Proposition 5.2 for the domain (−d, d) × (−l, l), and let

ηj = Iεj[vj, (−d, d) × (−l, l)] − 2dkh → 0 . (5.7)

By taking a subsequence we can assume the εj to be monotone. For each i,let j(i) be the smallest j > i such that εj < εi/i (this exists, since εj → 0).We rescale vj by εi/εj(i), to obtain

vi(x, y) =εiεj(i)

vj(i)

(

εj(i)εi

(x, y)

)

for |y| ≤ lεj(i)/εi, and continue it affinely for larger |y|. Scaling (5.7) gives

Iεi[vi, (−αid, αid) × R] ≤ 2αidkh + αiηj(i) ,

where αi = εi/εj(i) > i. It is clear that we can choose x0 such that

Iεi[vi, (x0 − d, x0 + d) × R] ≤ 2dkh + ηj(i) +

cd

i.

Hence the restriction of vi to (x0 − d, x0 + d)×R, translated to (−d, d)×R,satisfies (5.5)-(5.6), with Li = lαi.

37

Step 2. We now show that we find h > 0, L > 2h, δ > 0 (not dependingon i) and a sequence wi : (−d, d) × (−L,L) such that

lim supi→∞

Iεi[wi, (−d, d) × (−L,L)] ≤ 2dkh , (5.8)

with the additional properties that

for half of the y ∈ (L− h, L) , ess infx∈(−d,d)

|∇wsymi (x, y) −B| ≥ δ (5.9)

and, for some y2 = yi2 ∈ (−L,L− 2h),

for half of the y ∈ (y2, y2 + h) , ess infx∈(−d,d)

|∇wsymi (x, y) − A| ≥ δ (5.10)

(ess inf(·) ≥ δ means (·) ≥ δ a.e.).We choose δ < |A−B|/4, and define for y ∈ R and vi as in Step 1

fA(y) = H1 (x ∈ (−d, d) : |∇vsymi (x, y) − A| ≤ δ) ,

and analogously

fB(y) = H1 (x ∈ (−d, d) : |∇vsymi (x, y) −B| ≤ δ) .

Qualitatively, we expect fA(y) to equal 2d for large positive y and 0 for largenegative ones, with a small transition layer across the interface, and fB tobe essentially 2d − fA. If this is true, we can cut from the large (−Li, Li)domain in which vi is nonaffine a uniform region (of size 2L) where the sameholds, i.e. which still contains the interface.

In order to do this, first observe that since vi has energy bounded by2dkh + ηi ≤ c, the set of y such that fA(y) + fB(y) < 3d/2 has measure lessthan c1εi ≤ c1. Further, the set of y such that fA and fB are both nonzero hasmeasure less than c2, by Lemma 4.2(ii). It is also clear that fA(y) = 2d andfA(y) = 0 for y > Li and y < −Li, respectively. Further, if fA(y1) ≥ 3d/2and fB(y2) ≥ 3d/2, then Iεi

[vi, (−d, d)× (y1, y2)] ≥ cd, hence there can be atmost a finite number of such transitions (by (y1, y2) we mean the unorientedinterval whose endpoints are y1 and y2).

These results show that, away from a set γ of measure less than c1+c2, fAand fB are essentially characteristic functions of disjoint subsets. Precisely,we can decompose (−2Li, 2Li) into three disjoint subset α, β, γ such that

|γ| ≤ c1 + c2 ,

fA(y) ≥ 3d/2 , fB(y) = 0 if y ∈ α ,

fA(y) = 0 , fB(y) ≥ 3d/2 if y ∈ β ,

38

and the number of interfaces between α and β, i.e. of disjoint intervals (yk, y′k)

with yk ∈ α and y′k ∈ β, is bounded by c3.Consider the smallest y such that there is no point of β in a left h–

neighbourhood,y1 = inf y : (y − h, y) ∩ β = ∅ . (5.11)

We choose h > 2(c1+c2), so that (5.11) implies (5.9) in the interval (y1−h, y1).We now fix some large L (the precise value is given below), and consider theinterval (y1−L, y1−h). We divide it into sections of size h. By the definitionof y1, each of them intersects β. Now we show that, if L is large enough, thereis one that does not intersect α. Assume the contrary. Then in each sectionthere would be at least one interface between α and β. But the number ofinterfaces is less than c3, hence this is impossible provided that we chooseL > (c3 + 2)h. We conclude that there must be

y2 ∈ (y1 − L, y1 − 2h) such that (y2, y2 + h) ∩ α = ∅ .

This shows that the sequence wi(x, y) = vi(x, y−L+ y1) satisfies (5.8-5.10).

Step 3. To conclude the proof we show that we can restrict and translatethe sequence wi further, to obtain ui → u+

h in (−d, d) × (−l, l). Considerthe class of admissible limits, which are obtained from u+

h by translation andaddition of a constant and a skew-symmetric linear map, i.e.

G =

w0(x, y) = u+h (x, y − a) + S(x, y) + b

(5.12)

where a ∈ (−L+ h/2, L− h/2), S ∈ R2×2, Ssym = 0, and b ∈ R

2. We define

ηi = inf ‖wi − w0‖L1 : w0 ∈ G

as the distance to G, and claim that ηi → 0.If not, there would be a subsequence bounded from below, ηik > c > 0.

Consider the wik . By the compactness result (Theorem 2.1), there is a furthersubsequence which converges strongly in W 1,2 to some u0 as in Proposition2.2. Since the domain is a rectangle, u0 can have either only horizontal oronly vertical interfaces, but not both. By (5.9) one can show that it canhave no vertical interface, and therefore ∇usym

0 = A+ (B −A)g(y), for someg : (−l, l) → 0, 1. From (5.9) we further obtain that g(y) = 0 at leastfor half of the y in (L − h, L), and analogously from (5.10) we get thatg(y) = 1 at least for half of the y in (y2, y2 + h), hence u0 must contain somehorizontal interface. The only remaining case is that the limit has morethan one horizontal interface. But then, by the Γ-liminf result of Section3, the limiting energy would be at least 2ndkh, with n > 1, which is also

39

impossible. We conclude that the limit u0 has a single horizontal interface,located in (−L + h/2, L − h/2), and hence belongs to G. This contradictsthe assumption that ηik > c > 0.

The sequence ui is obtained from wi by elimination of the translation andthe rotation. For each i we choose wi

0 in G so that |wi − wi0| → 0, and set

ui(x, y) = wi(x, y + a) − S(x, y) − b

where a, S and b are as in (5.12), for wi0.

The following remark is a consequence of Proposition 5.5. It asserts thatProposition 5.2 remains valid also for prescribed sequences εi → 0.

Remark 5.6. Given d, l > 0 and a sequence εi → 0, there exists a sequenceui : (−d, d) → (−l, l) such that

limi→∞

Iεi[ui, (−d, d) × (−l, l)] = 2dkh , lim

i→∞‖ui − u+

h ‖L1 = 0 ,

which satisfies

ui(x, y) =

A(x, y)T if y ≥ l/2

(B + Sh + Si)(x, y)T + ci if y < −l/2 .

Proof. We define u′i as in Proposition 5.5 on the larger domain (−2d, 2d) ×(l, l), and then modify it as in Proposition 5.2.

6 Examples

The first term in the energy controls ε−1∫

min(|∇usym|2, |∇usym −B|2). It isnatural to ask if there is a rigidity argument, in the sense of a generalization ofKorn’s inequality to two wells, that gives directly control of ε−1

∫

|∇u−F |2,for some matrix F . We show here that this is not the case.

Remark 6.1. Let Ω = (−1, 1)2. For any sequence εi → 0, there is a sequenceui, such that ui = 0 on ∂Ω,

Iεi[ui,Ω] → 0 , ‖ui‖W 1,2 → 0 , and lim

i→∞inf

F∈R2×2

∫

Ω

1

εi|∇ui − F |2 → ∞ .

Proof. We write ui = ρεi∗ vi, where ρεi

is a mollification kernel on the scaleεi and vi is continuous and piecewise affine. Standard arguments show that

Iεi[ui,Ω] ≤ c

∫

Ω

1

εiW (∇vi) + c

∫

Ω

|∇2vi|

40

Figure 6.1: Subdivision of the domain for the construction of vi in Lemma6.1. The triangle PRQ is in the ’B’ phase, and has area hidi of order ε

1/2

i d2i .

The rest in the ’A’ phase.

where the second term has to be interpreted as the BV norm of ∇vi. Inthe following we drop for simplicity the index i. The construction of v is asshown in Figure 6.1. We set v = 0 outside the central quadrilateral, which hascorners P = (−d, 0), Q = (0,−h), R = (d, 0), S = (0, d). The quadrilateralPQRS is divided into four triangles, with one side in common with PQRSand one vertex in the origin. At the origin, we set v(0, 0) = (2hi, 0). In eachof the four triangles v is the linear interpolation between the values at thethree corners. Straightforward calculations show that

∫

Ω

W (∇v) ≤ ch2 ,

∫

Ω

|∇2v| ≤ cd ,

∫

Ω

|∇v|2 ≥ chd

for some universal c > 0. We conclude that for any sequence εi → 0, one canfind a sequence di which converges to zero slower than εi (e.g. di = ε

1/5i will

do) such that, setting hi = ε1/2i di, the thesis is satisfied.

It would be desirable to have a more explicit representation of the energydensity k(ν). This is possible in problems where the optimal profiles areone-dimensional. We show with an example that optimal profiles need notbe one-dimensional. The construction is a generalization to the case of linearelasticity of the one presented in [11] for the case of potentials W vanishingon two matrices, which in turn was inspired by an example from [18].

Remark 6.2. The two-well potential

W (∇u) =(

1 − (ux,y + uy,x − 1)2 + αux,xuy,y)2

+ u2x,x + u2

y,y ,

depends only on the symmetric part of the gradient and vanishes for A = 0and B = ex ⊗ ey + ey ⊗ ex. For α > 0 large, the optimal interfaces for the

41

energy functional

Iε[u] =

∫

Ω

1

εW (∇u) + ε|∇2u|2

are not one dimensional. More precisely, let u0(x, y) = (y + |y|, 0) and Ω =(−1, 1)2. Then there is no sequence uε(x, y) = uε(y) satisfying

Iε[uε,Ω] → I0[u0,Ω] , uε → u0 in L1(Ω).

Proof. In the class of one-dimensional interfaces the energy coincides withthe standard Modica-Mortola functional and is minimized by

u1dε (x, y) =

(

εv(y/ε)0

)

, v(y) = y + ln cosh y ,

and the limiting surface energy is k1d = 8/3. The perturbation

u2dε (x, y) =

(

εv(y/ε)0

)

+ λεψ(x

ε,y

ε

)

, ψ(x, y) =

(

F (y) sinxf(y) cos x

)

with F (y) and f(y) smooth functions converging to 0 as |y| → ∞, and λ > 0a small parameter clearly satisfy u2d

ε → u0 as ε→ 0. Expanding Iε in λ yieldsthat for F (y) = 1/ cosh(y) and with α large enough, the limiting energy ofu2dε is strictly smaller than that of u1d

ε .

We note that W (F ) is invariant under swapping the x and y coordinates,and hence the same result holds for k(ex). Application of a straightforwardtruncation to the first term in W permits to obtain a potential W ∗ withquadratic growth at infinity such thatW (F ) = W ∗(F ) whenever |F sym| ≤M ,for any large M . The remark remains valid with only notational changes.

Acknowledgements

The main part of this work was carried out while the first author was affili-ated with, and the second author was visiting, the Max-Planck-Institute forMathematics in the Sciences in Leipzig, Germany. The second author wishesto thank the MPI for the kind hospitality. The first author was partiallysupported by the DFG Schwerpunktprogramm 1095 Analysis, Modeling andSimulation of Multiscale Problems. We thank Miroslav Chlebik for bringingreference [21] to our attention, and Peter Hornung for useful comments on afirst version of this paper.

42

References

[1] L. Ambrosio, Metric space valued functions of bounded variation, Ann.Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 439–478.

[2] L. Ambrosio, C. DeLellis, and C. Mantegazza, Line energies for gradientvector fields in the plane, Calc. Var. Part. Diff. Eq. 9 (1999), 327–355.

[3] S. Baldo, Minimal interface criterion for phase transitions in mixtures ofCahn-Hilliard fluids, Ann. Inst. H. Poicare Anal. Non Lineare 7 (1990),67–90.

[4] J. Ball, A version of the fundamental theorem for Young measures,PDE’s and Continuum Models of Phase Transitions (M. Rascle, D. Serre,and M. Slemrod, eds.), Lecture Notes in Physics, vol. 344, Springer-Verlag, Berlin, 1989, pp. 207–215.

[5] J. Ball and R. D. James, Fine phase mixtures as minimizers of theenergy, Arch. Rat. Mech. Anal. 100 (1987), 13–52.

[6] A. C. Barroso and I. Fonseca, Anisotropic singular perturbations - thevectorial case, Proc. Roy. Soc. Edin. Sect. A 124 (1994), 527–571.

[7] K. Bhattacharya, Microstructure of martensite: Why it forms and howit gives rise to the shape-memory effect, Oxford University Press, 2003.

[8] G. Bouchitte, Singular perturbations of variational problems arising froma two-phase transition model, Appl. Math. Optim. 21 (1990), 289–314.

[9] A. Braides, Γ-convergence for beginners, Oxford University Press, Ox-ford, 2002.

[10] A. Braides and A. Defranceschi, Homogenization of multiple integrals,Claredon Press, Oxford, 1998.

[11] S. Conti, I. Fonseca, and G. Leoni, A Γ–convergence result for thetwo-gradient theory of phase transitions, Comm. Pure Appl. Math. 55(2002), 857–936.

[12] G. Dal Maso, An introduction to Γ-convergence, Birkhauser, Boston,1993.

[13] A. DeSimone, R. V. Kohn, S. Muller, and F. Otto, A compactness resultin the gradient theory of phase transitions, Proc. Roy. Soc. Edin. A 131(2001), 833–844.

43

[14] G. Dolzmann and S. Muller, Microstructures with finite surface energy:the two-well problem, Arch. Rat. Mech. Anal. 132 (1995), 101–141.

[15] I. Fonseca and C. Mantegazza, Second order singular perturbation modelsfor phase transitions, SIAM J. Math. Anal. 31 (2000), no. 5, 1121–1143.

[16] I. Fonseca and L. Tartar, The gradient theory of phase transitions forsystems with two potential wells, Proc. Roy. Soc. Edin. Sect. A 111(1989), 89–102.

[17] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale,Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. (8) (1975), 842–850.

[18] W. Jin and R. V. Kohn, Singular perturbation and the energy of folds,J. Nonlinear Sci. 10 (2000), 355–390.

[19] R. V. Kohn and S. Muller, Surface energy and microstructure in coherentphase transitions, Comm. Pure Appl. Math. 47 (1994), 405–435.

[20] R. V. Kohn and P. Sternberg, Local minimizers and singular perturba-tions, Proc. Roy. Soc. Edin. Sect. A 111 (1989), 69–84.

[21] P. Komjath, Large sets not containing images of a given sequence,Canad. Math. Bull. 26 (1983), 41–43.

[22] C. De Lellis and F. Otto, Structure of entropy solutions to the eikonalequation, J. Eur. Math. Soc. 5 (2003), 107–145.

[23] L. Modica and S. Mortola, Un esempio di Γ-convergenza, Boll. Un. Mat.Ital. (5) 14-B (1977), 285–299.

[24] S. Muller, Variational models for microstructure and phase transitions,in: Calculus of variations and geometric evolution problems (F. Bethuelet al., eds.), Springer Lecture Notes in Math. 1713, Springer, Berlin,1999, pp. 85–210.

[25] N. C. Owen and P. Sternberg, Nonconvex variational problems withanisotropic perturbations, Nonlin. Anal. 16 (1991), 705–719.

[26] P. Sternberg, Vector-valued local minimizers of nonconvex variationalproblems, Rocky Mountain J. Math. 21 (1991), 799–807.

[27] E. Zeidler, Nonlinear functional analysis and its applications, vol. IV,Springer, 1997.

44