Milano, 19 January 2017 - aim-mate.it · 2017. 1. 28. · Marco Bramanti (Politecnico di Milano) Hörmander operators Milano, 19 January 2017 5 / 15. Degenerate elliptic equations

Hörmander operators

Marco Bramanti

Politecnico di Milano

Milano, 19 January 2017

Marco Bramanti (Politecnico di Milano) Hörmander operators Milano, 19 January 2017 1 / 15

Introduction

The aim of this presentation is to give a first outline of an active fieldof research which lies in the area of the theoretical study of partialdifferential equations (PDEs).

This field can also be the object of M.Sc. or Ph.D. thesis. However,in this presentation I will not describe specific problems which can bethe subject of a thesis.

I will just try to communicate some elements of this field of researchthat can be interesting, while specific research problems can beappreciated only after some preliminary work aimed to deepening theknolwledge of the background.

“Theoretical study of partial differential equations”means that thecore of the work is to prove new theorems related to the properties ofsolutions of certain PDEs.


Introduction






Introduction






Introduction






Regularity of solutions of elliptic and parabolic equations

Let us focus on linear second order PDEs. The three most famousequations of this kind are (exemplifying equations in 3 variables):

the Laplace (or Poisson) equation: ∆u = uxx + uyy + uzz = fthe diffusion equation: Hu ≡ ut − k∆u = fthe wave equation: utt − c2∆u = fIt is well know that solutions to the first two equations are smooth(i.e., infinitely differentiable) as soon as f = 0 or f is a smoothfunction, while the wave equation can have, even for f = 0,nonsmooth solutions.




the Laplace (or Poisson) equation: ∆u = uxx + uyy + uzz = f

the diffusion equation: Hu ≡ ut − k∆u = fthe wave equation: utt − c2∆u = fIt is well know that solutions to the first two equations are smooth(i.e., infinitely differentiable) as soon as f = 0 or f is a smoothfunction, while the wave equation can have, even for f = 0,nonsmooth solutions.




the Laplace (or Poisson) equation: ∆u = uxx + uyy + uzz = fthe diffusion equation: Hu ≡ ut − k∆u = f

the wave equation: utt − c2∆u = fIt is well know that solutions to the first two equations are smooth(i.e., infinitely differentiable) as soon as f = 0 or f is a smoothfunction, while the wave equation can have, even for f = 0,nonsmooth solutions.




the Laplace (or Poisson) equation: ∆u = uxx + uyy + uzz = fthe diffusion equation: Hu ≡ ut − k∆u = fthe wave equation: utt − c2∆u = f

It is well know that solutions to the first two equations are smooth(i.e., infinitely differentiable) as soon as f = 0 or f is a smoothfunction, while the wave equation can have, even for f = 0,nonsmooth solutions.




the Laplace (or Poisson) equation: ∆u = uxx + uyy + uzz = fthe diffusion equation: Hu ≡ ut − k∆u = fthe wave equation: utt − c2∆u = fIt is well know that solutions to the first two equations are smooth(i.e., infinitely differentiable) as soon as f = 0 or f is a smoothfunction, while the wave equation can have, even for f = 0,nonsmooth solutions.


Regularity of solutions of elliptic and parabolic equationsThe Poisson and the diffusion equation are prototypes of more generalelliptic or parabolic equations (in n or n+ 1 variables):

Lu ≡n

∑i ,j=1

aij (x) uxi xj = f (x) (1)

ut − Lu ≡ ut −n

∑i ,j=1

aij (x , t) uxi xj = f (x , t)

If the matrix aij (x)ni ,j=1 is positive definite, uniformly in thedomain Ω where the equation is studied:

n

∑i ,j=1

aij (x) ξ i ξ j ≥ λ |ξ|2

for some constant λ > 0, every ξ ∈ Rn and x ∈ Ω, then we say thatthese equations are uniformly elliptic or parabolic, respectively.Uniformly elliptic or parabolic equations are strongly regularizing:solutions to Lu = f or ut − Lu = f are smooth as soon as f issmooth.



Lu ≡n

∑i ,j=1



∑i ,j=1



n

∑i ,j=1


for some constant λ > 0, every ξ ∈ Rn and x ∈ Ω, then we say thatthese equations are uniformly elliptic or parabolic, respectively.

Uniformly elliptic or parabolic equations are strongly regularizing:solutions to Lu = f or ut − Lu = f are smooth as soon as f issmooth.



Lu ≡n

∑i ,j=1



∑i ,j=1



n

∑i ,j=1


for some constant λ > 0, every ξ ∈ Rn and x ∈ Ω, then we say thatthese equations are uniformly elliptic or parabolic, respectively.Uniformly elliptic or parabolic equations are strongly regularizing:solutions to Lu = f or ut − Lu = f are smooth as soon as f issmooth.


Degenerate elliptic equations

We say that an equation (1) is degenerate elliptic if the matrixaij (x)ni ,j=1 is nonnegative definite, but is not uniformly positivedefinite.

Solutions to degenerate elliptic equations can be nonsmooth: forinstance, if in R3 we look for a solution u (x , y , z) to

uxx + uyy = 0

then every u (x , y , z) = g (z) (even discontinuous!) will go.

An interesting phenomenum that may occur is the following.

There exist some degenerate elliptic equations which are, nevertheless,strongly regularizing, due to the special form of their matrix.





uxx + uyy = 0








uxx + uyy = 0








uxx + uyy = 0





An example of degenerate elliptic but hypoelliptic operatorConsider, in R3, the equation:

Lu ≡[(∂x + y∂z )

2 + (∂y − x∂z )2]u (x , y , z) = f (x , y , z) .

Expanding the formal squares:

uxx + uyy +(x2 + y2

)uzz + 2 (yuxz − xuyz ) = f

A =

1 0 −x0 1 y−x y x2 + y2

; detA ≡ 0which shows that the equation degenerates everywhere.Nevertheless, it can be proved that the operator L is hypoelliptic, i.e.:if the equation Lu = f is satisfied (also in distributional sense) inΩ ⊂ R3 then

f ∈ C∞ (Ω) =⇒ u ∈ C∞ (Ω) .



Lu ≡[(∂x + y∂z )

2 + (∂y − x∂z )2]u (x , y , z) = f (x , y , z) .


uxx + uyy +(x2 + y2


A =

1 0 −x0 1 y−x y x2 + y2

; detA ≡ 0which shows that the equation degenerates everywhere.

Nevertheless, it can be proved that the operator L is hypoelliptic, i.e.:if the equation Lu = f is satisfied (also in distributional sense) inΩ ⊂ R3 then

f ∈ C∞ (Ω) =⇒ u ∈ C∞ (Ω) .



Lu ≡[(∂x + y∂z )

2 + (∂y − x∂z )2]u (x , y , z) = f (x , y , z) .


uxx + uyy +(x2 + y2


A =

1 0 −x0 1 y−x y x2 + y2

; detA ≡ 0which shows that the equation degenerates everywhere.Nevertheless, it can be proved that the operator L is hypoelliptic, i.e.:

if the equation Lu = f is satisfied (also in distributional sense) inΩ ⊂ R3 then

f ∈ C∞ (Ω) =⇒ u ∈ C∞ (Ω) .



Lu ≡[(∂x + y∂z )

2 + (∂y − x∂z )2]u (x , y , z) = f (x , y , z) .


uxx + uyy +(x2 + y2


A =

1 0 −x0 1 y−x y x2 + y2

; detA ≡ 0which shows that the equation degenerates everywhere.Nevertheless, it can be proved that the operator L is hypoelliptic, i.e.:if the equation Lu = f is satisfied (also in distributional sense) inΩ ⊂ R3 then

f ∈ C∞ (Ω) =⇒ u ∈ C∞ (Ω) .


Commutators of vector fieldsTo better understand this phenomenum, let us consider again theoperator

L ≡ X 2 + Y 2 ≡ (∂x + y∂z )2 + (∂y − x∂z )

2

Geometrically, the operator L is degenerate because it is built in sucha way to control only 2 directions in R3, those of the vector fields

X = ∂x + y∂z = (1, 0, y)

Y = ∂y − x∂z = (0, 1,−x) .

However, let us compute

XY = (∂x + y∂z ) (∂y − x∂z ) = ∂2xy + y∂2yz − ∂z − x∂2xz − xy∂2zz

YX = (∂y − x∂z ) (∂x + y∂z ) = ∂2xy − x∂2xz + ∂z + y∂2yz − xy∂2zz

[X ,Y ] ≡ XY − YX = −2∂z

The new vector field [X ,Y ] ≡ XY − YX is called the commutator ofX and Y .



L ≡ X 2 + Y 2 ≡ (∂x + y∂z )2 + (∂y − x∂z )

2


X = ∂x + y∂z = (1, 0, y)

Y = ∂y − x∂z = (0, 1,−x) .




[X ,Y ] ≡ XY − YX = −2∂z




L ≡ X 2 + Y 2 ≡ (∂x + y∂z )2 + (∂y − x∂z )

2


X = ∂x + y∂z = (1, 0, y)

Y = ∂y − x∂z = (0, 1,−x) .




[X ,Y ] ≡ XY − YX = −2∂z




L ≡ X 2 + Y 2 ≡ (∂x + y∂z )2 + (∂y − x∂z )

2


X = ∂x + y∂z = (1, 0, y)

Y = ∂y − x∂z = (0, 1,−x) .




[X ,Y ] ≡ XY − YX = −2∂z



Hörmander’s condition and Hörmander’s theorem

Note that the three directions

X = ∂x + y∂z = (1, 0, y)Y = ∂y − x∂z = (0, 1,−x)[X ,Y ] = −2∂z = (0, 0,−2)

are linearly independent at every point of R3.

In some sense, the third direction of R3 which is missing in theoperator L ≡ X 2 + Y 2 is recovered by the commutator [X ,Y ] of thetwo vector fields X ,Y which constitute L.

This fact is deeply related to the regularizing properties of theoperator L, as explained by the next fundamental result.




X = ∂x + y∂z = (1, 0, y)Y = ∂y − x∂z = (0, 1,−x)[X ,Y ] = −2∂z = (0, 0,−2)







X = ∂x + y∂z = (1, 0, y)Y = ∂y − x∂z = (0, 1,−x)[X ,Y ] = −2∂z = (0, 0,−2)





Theorem (Hörmander, 1967)Let L be a linear second order partial differential operator with real smoothcoeffi cients which can be written as

L =q

∑i=1X 2i + X0, with Xi =

n

∑j=1bij (x) ∂xj (2)

for x ∈ Ω ⊂ Rn. (For q < n it is degenerate elliptic; for q + 1 < n it isalso degenerate parabolic).Assume that the vector fields X0,X1, ...,Xq satisfy Hörmander’s condition,that is: among the vector fields Xi , their commutators [Xi ,Xj ], theiriterated commutators [[Xi ,Xj ] ,Xk ] and so on, there exist, at every pointof Ω, n independent vectors.Then the operator L is hypoelliptic in Ω: Lu ∈ C∞ (Ω) =⇒ u ∈ C∞ (Ω).

Operators of type (2) with the vector fields Xi satisfying Hörmander’scondition are called Hörmander operators.


The theory of Hörmander operators

The theory of Hörmander operators has by now a history of 50 years,and is still an active field of research.

The general phylosophy of the theory is that these operators exhibit“good properties” similar to those of elliptic opeators (and muchbetter than those of general degenerate operators), provided one findsthe right analytical / geometric / algebraic way to formulate theseproperties.


The theory of Hörmander operators

The theory of Hörmander operators has by now a history of 50 years,and is still an active field of research.

The general phylosophy of the theory is that these operators exhibit“good properties” similar to those of elliptic opeators (and muchbetter than those of general degenerate operators), provided one findsthe right analytical / geometric / algebraic way to formulate theseproperties.


Motivation and examples

There are important problems in analysis and applied mathematicswhich lead to the study of partial differential operators of Hörmandertype.

A source of examples of this kind is that ofKolmogorov-Fokker-Planck equations which describe the evolution ofa system with a finite number of degrees of freedom, subject to adeterministic dynamic but also perturbed by a random component(white noise):

in this case the probability transition density p obeys an equation ofthe kind

∂tp +n

∑j=1bj (x) ∂xjp +

n

∑i ,j=1

aij (x) ∂2xi xjp = 0

where the matrix aij is often lower dimensional (degenerate parabolic).






∂tp +n


n

∑i ,j=1








∂tp +n


n

∑i ,j=1





Examples of Hörmander operators which are particular instances ofKolmogorov-Fokker-Planck equations are:

the model of a particle moving of Brownian motion;

the Mumford operator which arises in the study of the “process ofrandom direction”;

the August-Zucker model to study the “process of random curvature”(both used in image processing);

the Citti-Sarti model in neurogeometry of human vision...






























Some ideas of the theory. Translations, dilations,fundamental solutions

A special class of Hörmander operators consists in operators whichare translation invariant with respect to a suitable (non-Euclidean)group of translations in Rn and 2-homogeneous with respect to a(nonisotropic) family of dilations (as the classical Laplacian is w.r.t.Euclidean translations and dilations).

These operators L possesses a fundamental solution in Rn of the kind

ΓL (x) =c

‖x‖Q−2(analogous to: Γ∆ (x) =

c

|x |n−2)

where ‖·‖ is a homogeneous norm with respect to the dilations, Q isthe “homogeneous dimension”. Then for every u ∈ C∞

0 (Rn)

u (x) =∫

ΓL(y−1 x

)Lu (y) dy = L

∫ΓL(y−1 x

)u (y) dy

where is the group operation (“translation”).This is a classical fundamental result of the theory, and a useful toolfor proving many subsequent results.



A special class of Hörmander operators consists in operators whichare translation invariant with respect to a suitable (non-Euclidean)group of translations in Rn and 2-homogeneous with respect to a(nonisotropic) family of dilations (as the classical Laplacian is w.r.t.Euclidean translations and dilations).These operators L possesses a fundamental solution in Rn of the kind

ΓL (x) =c


c

|x |n−2)


0 (Rn)

u (x) =∫

ΓL(y−1 x

)Lu (y) dy = L

∫ΓL(y−1 x

)u (y) dy

where is the group operation (“translation”).

This is a classical fundamental result of the theory, and a useful toolfor proving many subsequent results.



A special class of Hörmander operators consists in operators whichare translation invariant with respect to a suitable (non-Euclidean)group of translations in Rn and 2-homogeneous with respect to a(nonisotropic) family of dilations (as the classical Laplacian is w.r.t.Euclidean translations and dilations).These operators L possesses a fundamental solution in Rn of the kind

ΓL (x) =c


c

|x |n−2)


0 (Rn)

u (x) =∫

ΓL(y−1 x

)Lu (y) dy = L

∫ΓL(y−1 x

)u (y) dy

where is the group operation (“translation”).This is a classical fundamental result of the theory, and a useful toolfor proving many subsequent results.


Some ideas of the theory. Sobolev spaces and a prioriestimates

For Hörmander operators

Lu =q

∑i=1X 2i u + X0u,

one can prove (local) a priori estimates hold on the second derivativeswith respect to the vector fields involved in the operator:q

∑i ,j=1‖XiXju‖Lp (Ω′) +

q

∑i=0‖Xiu‖Lp (Ω′) ≤ c

‖Lu‖Lp (Ω) + ‖u‖Lp (Ω)

,

for 1 < p < ∞ and Ω′ b Ω.

This is the analog of classical a priori estimates for elliptic operators:n

∑i ,j=1

∥∥uxi xj∥∥Lp (Ω) + n

∑i=1‖uxi ‖Lp (Ω) ≤ c


This means that the standard Sobolev spaces must be replaced, inthis context, by those induced by the vector fields.




Lu =q

∑i=1X 2i u + X0u,



q



,

for 1 < p < ∞ and Ω′ b Ω.This is the analog of classical a priori estimates for elliptic operators:

n

∑i ,j=1








Lu =q

∑i=1X 2i u + X0u,



q



,

for 1 < p < ∞ and Ω′ b Ω.This is the analog of classical a priori estimates for elliptic operators:

n

∑i ,j=1






Concluding remarks

This area is still active and presents a number of open problems,sometimes highly challenging. It can also offer the material fordifferent kinds of thesis. Besides the possible original content, I thinkthat preparing a thesis in this area is surely an instructive experience,for the variety and depth of mathematical tools and ideas that areinvolved (from PDEs, geometry, algebra...).

For the interested student a suggested reading is the survey booklet:

M. Bramanti: An invitation to hypoelliptic operators and Hörmander’svector fields. Springer Briefs in Mathematics. (Biblioteca del Dip. diMatematica, Biblioteca Centrale di Ingegneria).


Concluding remarks





Concluding remarks





Milano, 19 January 2017 - aim-mate.it · 2017. 1. 28. · Marco Bramanti (Politecnico di Milano) Hörmander operators Milano, 19 January 2017 5 / 15. Degenerate elliptic equations

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