symmetry and spectral properties for viscosity solutions

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

Symmetry and spectral properties for viscositysolutions of fully nonlinear equations

Isabeau Birindelli

Sapienza Universita di Roma

Nonlinear PDEs: Optimal Control, Asymptotic Problems and MeanField Games, Padova 25 febbraio 2016,

Joint work with Fabiana Leoni and Filomena Pacella.


GOAL Obtain symmetry properties ofsolutions of fully nonlinear equations related to spectral propertiesof what, improperly, will be called the linearized operator.


Introduction

GOAL Obtain symmetry properties of solutions of fully nonlinearequations related to spectral properties of what, improperly, will becalled the linearized operator.Question: which symmetry features of the domain and theoperator are inherited by the viscosity solutions of thehomogeneous Dirichlet problem

−F (x ,D2u) = f (x , u) in Ω ,

u = 0 on ∂Ω ,(1)

where Ω ⊂ IRn, n ≥ 2, is a bounded domain and F is a fullynonlinear uniformly elliptic operator?


Introduction

We are under the condition that F is Lipschitz continuous in x anduniformly elliptic i.e. ∀ x ∈ Ω

M−α,β(M−N) ≤ F (x ,M)−F (x ,N) ≤M+α,β(M−N) , M,N ∈ Sn ,

where M−α,β and M+α,β are the Pucci’s extremal operators with

ellipticity constants 0 < α ≤ β.i.e.

M+α,β(M) = sup

αI≤A≤βItrAM = α

∑ei<0

ei (M) + β∑ei>0

ei (M)

M−α,β(M) = infαI≤A≤βI

trAM = β∑ei<0

ei (M) + α∑ei>0

ei (M)


Introduction

Invariance with respect to reflection

For a given unit vector e, let H(e) be the plane orthogonal to eand σ(e) the reflection with respect to e . F is invariant withrespect to the reflection σ(e) if

F (σe(x), (I − 2e ⊗ e)M(I − 2e ⊗ e)) = F (x ,M) ∀ x ∈ Ω , M ∈ Sn

F is invariant with respect to rotation if

F(Ox ,OMOT

)= F (x ,M) ∀ x ∈ Ω , M ∈ Sn all orthogonal matrixO

Observe that M and (In − 2e ⊗ e)M(In − 2e ⊗ e) and OMOT havethe same eigenvalues. So any operator that depends only on theeigenvalues of the Hessian is invariant with respect to reflectionand to rotation e.g. Pucci’s operators, the Laplace operator,Monge Ampere....


Introduction

Eigenfunctions of the Laplacian in the ball

∆u + λu = 0 in B ,

u = 0 on ∂B ,

It is well known that the principal eigenfunction (which is positive)is radial.


Introduction

Eigenfunctions of the Laplacian in the ball

On the other hand, the eigenfunctions corresponding to thesecond eigenvalue are not radial, they are equal to

φ2(x) =x · e|x |

J(|x |).

In particular, we cannot expect general solutions to inherit ”all thesymmetries” of the domain and the operator.


Introduction

Starting with Alexandrov and after the fundamental works ofSerrin and Gidas, Ni, Nirenberg most results on symmetry ofsolutions rely on the moving plane method. It is impossible to evenstart mentioning all the results obtained via that method, be theyfor semilinear, quasilinear or fully nonlinear equations.For positive solutions of fully nonlinear equations: Bardi (1985),Badiale, Bardi (1990),Da Lio and Sirakov , B. and Demengel,Silvestre and Sirakov .


Introduction

The much acclaimed moving plane method is the tool that allowsto extend the symmetry of the principal eigenfunction to positivesolutions of semi-linear equations.

−∆u = f (x , u) in Ω ,

u = 0 on ∂Ω .

Can the analogy be continued? when can one expect solutions ofnonlinear equations to share the same symmetry of othereigenfunctions?which is the right symmetry to consider?


Introduction


−∆u = f (x , u) in Ω ,

u = 0 on ∂Ω .

Can the analogy be continued?

when can one expect solutions ofnonlinear equations to share the same symmetry of othereigenfunctions?which is the right symmetry to consider?


Introduction


−∆u = f (x , u) in Ω ,

u = 0 on ∂Ω .

Can the analogy be continued? when can one expect solutions ofnonlinear equations to share the same symmetry of othereigenfunctions?

which is the right symmetry to consider?


Introduction


−∆u = f (x , u) in Ω ,

u = 0 on ∂Ω .

Can the analogy be continued? when can one expect solutions ofnonlinear equations to share the same symmetry of othereigenfunctions?which is the right symmetry to consider?


Introduction

Limits of applications of the moving plane method.

−∆u = f (x , u) in Ω ,

u = 0 on ∂Ω .

Sign changing solutions

Non convex domains e.g. when Ω is an annulus.

If the nonlinear term f (x , u) does not have the rightmonotonicity in the x variable.


Introduction

Foliated Schwarz Symmetry

Definition

Let B be a ball or an annulus in IRn. A function u : B → IR isfoliated Schwarz symmetric if there exists a unit vector p ∈ Sn−1

such that u(x) only depends on |x | and θ = arccos(

x|x | · p

), and u

is non increasing with respect to θ ∈ (0, π) .

Though more telling would be to call it axial Schwarz symmetry.


Introduction

Lemma

u is foliated Schwarz symmetric with respect to the directionp ∈ Sn−1 if and only if u(x) ≥ u(σe(x)) for all x ∈ B(e) and forevery e ∈ Sn−1 such that e · p ≥ 0.

Whose consequence is

Proposition

A function u ∈ C 1(B) ∩ C (B) is foliated Schwarz symmetric if andonly if there exists a direction e ∈ Sn−1 such that u is symmetricwith respect to H(e) and for any other direction e ′ ∈ Sn−1 \ ±eone has either uθe,e′ ≥ 0 in B(e) or uθe,e′ ≤ 0 in B(e).


Introduction

In the last decades, some work has been devoted to understandingunder which conditions solutions of semilinear elliptic equations arefoliated Schwarz symmetric. This line of research, which stronglyrelies on the maximum principle, was started by Pacella and thendeveloped by Pacella and Weth. See also-Gladiali, Pacella,Weth-Pacella, Ramaswamy-Weth .


Introduction

In order to prove foliated Schwarz symmetry of a solution u of∆u + f (|x |, u) = 0 in B ,

u = 0 on ∂B .

one needs to study the sign of the first eigenvalue λ1(Lu,B(e)) ofthe linearized operator Lu = ∆ + ∂f

∂u (|x |, u) at the solution u, in thehalf domain B(e) = x ∈ B : x · e > 0 for a direction e ∈ Sn−1.

Question: what plays the role of the linearized operator for thefully nonlinear problem?


Introduction

M−α,β(M−N) ≤ F (x ,M)−F (x ,N) ≤M+α,β(M−N) , M,N ∈ Sn ,

This, in order, will imply that if u is a solution of−F (x ,D2u) = f (x , u) in Ω ,

u = 0 on ∂Ω ,

any v ”derivative of u ” will satisfy

−M+α,β(D2v) ≤ ∂f

∂u(x , u) v in Ω ,

and

−M−α,β(D2v) ≥ ∂f

∂u(x , u) v in Ω .

v being only a viscosity subsolution or supersolution.


Introduction

The fully nonlinear operator

Lu(v) : =M+α,β(D2v) +

∂f

∂u(x , u) v .

will be improperly called ”linearized operator at u”.

Proposition

Assume F is invariant with respect to rotation, that Ω = B isradially symmetric and λ+

1 (Lu,B) > 0 then u is radial.

Proposition

Assume that Ω and F are symmetric with respect to thehyperplane H(e), let Ω(e) = Ω ∩ x · e ≥ 0

(i) If f is convex and λ+1 (Lu,Ω(±e)) > 0

or

(ii) If f is strictly convex and λ+1 (Lu,Ω(±e)) ≥ 0

Then, u is symmetric with respect to the hyperplane H(e).


Introduction

Definition of the eigenvalue.

Following the ideas of Berestycki, Nirenberg, Varadhan,

let

λ+1 (F ,Ω) := supµ ∈ IR, ∃ φ > 0 in Ω, F [φ] + µφ ≤ 0 .

λ−1 (F ,Ω) := supµ ∈ IR, ∃ φ < 0 in Ω, F [φ] + µφ ≥ 0 .

with e.g. F [φ] =M+α,β(D2φ) + c(x)φ.


Introduction

There exists φ > 0 (and ψ < 0) in Ω such thatF [φ] + λ+

1 (F ,Ω)φ = 0 in Ωφ = 0 on ∂ΩF [ψ] + λ−1 (F ,Ω)ψ = 0 in Ωψ = 0 on ∂Ω

in the viscosity sense. (See the works of Lions,Busca-Esteban-Quaas, Ishii-Yoshimura, B.-Demengel,Sirakov-Quaas, Armstrong,...)


Introduction

Proposition

With the above notations, the following properties hold:

(i) If D1 ⊂ D2 and D1 6= D2, then λ±1 (D1) > λ±1 (D2) .

(ii) For a sequence of domains Dk such that Dk ⊂ Dk+1, then

limk→+∞

λ±1 (Dk) = λ±1 (∪kDk) .

(iii) If F [φ] =M+α,β(D2φ) + c(x)φ and α < β then λ+

1 < λ−1 .

(iv) If λ 6= λ±1 is an eigenvalue then every correspondingeigenfunction changes sign.

(v) λ+1 (D) > 0 (λ−1 (D) > 0) if and only if the maximum

(minimum) principle holds for F in D .

(vi) λ±1 (D)→ +∞ as meas(D)→ 0 .


Introduction

Proposition

Assume that F is invariant with respect to rotation, Ω = B isradially symmetric and λ+

1 (Lu,B) > 0 then u is radial.

Proposition

Assume that Ω and F are symmetric with respect to thehyperplane H(e),

(i) If f is convex and λ+1 (Lu,Ω(±e)) > 0

or

(ii) If f is strictly convex and λ+1 (Lu,Ω(±e)) ≥ 0

Then, u is symmetric with respect to the hyperplane H(e).

Proof: (On the board)


Foliated Schwarz symmetry

Theorem

Suppose that F is invariant with respect to any reflection σe andby rotations. Let u be a viscosity solution of problem

−F (x ,D2u) = f (x , u) in B ,

u = 0 on ∂B ,

with f (x , ·) = f (|x |, ·) convex in IR. If there exists e ∈ Sn−1 suchthat

λ+1 (Lu,B(e)) ≥ 0,

then u is foliated Schwarz symmetric.


Foliated Schwarz symmetry

On the other hand one has the following necessary condition.

Theorem

Assume that the solution u is not radial but it is foliated Schwarzsymmetric with respect to p ∈ Sn−1. Then, for all e ∈ Sn−1 suchthat e · p = 0, one has

λ−1 (Lu,B(e)) ≥ 0 .

Proposition

Let u ∈ C 1(Ω) be a sign changing viscosity solution, with Finvariant with respect to reflection to the hyperplane H(e) (saye = e1) and assume that u is even with respect to x1. Then

λ+1 (Lu,Ω(±e1)) ≥ 0 =⇒ N (u) ∩ ∂Ω 6= ∅ .

where N (u) is the nodal set of u.


Spectral properties

In any bounded domain Ω, one can define

µ+2 (Lu,Ω) = inf

D⊂Ωmax

λ+

1 (Lu,D), λ+1 (Lu,Ω \ D)

(2)

where the infimum is taken on all subdomains D contained in Ω.When Lu = ∆ + f ′(|x |, u), µ+

2 = Λ2 i.e. it is just the secondeigenvalue of Lu.

Corollary

Under the above assumptions, if µ+2 (Lu,Ω) ≥ 0 then u is foliated

Schwarz symmetric.

It turns out that:

Proposition

If α < β, then µ+2 (Lu,Ω) is not an eigenvalue for Lu in Ω with

corresponding sign changing eigenfunctions having exactly twonodal regions.

Proof: (On board)


Nodal eigenvalues

This Proposition leads to believe that a natural candidate for beingthe second eigenvalue of Lu could be

γ+2 (Lu,B) = inf

D⊂Bmax

λ+

1 (Lu,D), λ−1 (Lu,B \ D)≥ µ+

2 (Lu,B) .

It would be also interesting to know, at least, whether the nonnegativity of γ+

2 (Lu,B) implies that u be foliated Schwarzsymmetric.


Nodal eigenvalues

For F uniformly elliptic and positively one homogeneous operatorlet

Λ2(F ) = infλ > maxλ−1 (F ), λ+1 (F ) : λ is an eigenvalue of F .

(3)It was proved by Armstrong, that Λ2(F ) > maxλ−1 (F ), λ+

1 (F )and that for any

µ ∈ (maxλ−1 (F ), λ+1 (F ),Λ2(F ))

and, for any continuous f , there exists a solution of the Dirichletproblem

F (x ,D2u) + µu = f (x) in Bu = 0 on ∂B.


Nodal eigenvalues

The next result relates the principal eigenvalue of M+α,β in any half

domain B(e) i.e. λ+1 (M+

α,β,B(e)) with Λ2(F ,B) and λr2(F ,B)which denotes the smallest radial nodal eigenvalue of F in B.

Theorem

Let F be positively one homogeneous, then the followinginequalities hold

λr2(F ,B) > λ+1 (M+

α,β,B(e)) and Λ2(F ,B) ≥ λ+1 (M+

α,β,B(e)) .


Nodal eigenvalues

To conclude, we observe that an important question which remainsopen is whether λ+

1 (M+α,β,B(e)) is a nodal eigenvalue for M+

α,β inB, as for the laplacian, or not.

Proposition

Assume that λ+1 (M+

α,β,B(e)) is a nodal eigenvalue for M+α,β in B

and that ψ2 is a corresponding eigenfunction, i.e.M+

α,β(D2ψ2) + λ+1 (M+

α,β,B(e))ψ2 = 0 in B

ψ2 = 0 on ∂B

Then

(i) ψ2 is not radial;

(ii) ψ2 is foliated Schwarz symmetric;

(iii) the nodal set of N (ψ2) does intersect the boundary;

(iv) if α < β, then, for any e ∈ Sn−1,B+ := x ∈ B : ψ2 > 0 6= B(e).


Nodal eigenvalues

symmetry and spectral properties for viscosity solutions

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