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Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

Isabeau Birindelli

Sapienza Universit` a di Roma

Nonlinear PDEs: Optimal Control, Asymptotic Problems and Mean Field Games, Padova 25 febbraio 2016, Joint work with Fabiana Leoni and Filomena Pacella.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations

GOAL Obtain symmetry properties of solutions of fully nonlinear equations related to spectral properties

• f what, improperly, will be called the linearized operator.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

GOAL Obtain symmetry properties of solutions of fully nonlinear equations related to spectral properties of what, improperly, will be called the linearized operator. Question: which symmetry features of the domain and the

• perator are inherited by the viscosity solutions of the

homogeneous Dirichlet problem

• −F(x, D2u) = f (x, u)

in Ω , u = 0

• n ∂Ω ,

(1) where Ω ⊂ I Rn, n ≥ 2, is a bounded domain and F is a fully nonlinear uniformly elliptic operator?

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

We are under the condition that F is Lipschitz continuous in x and uniformly 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≤βI

trAM = α

• ei<0

ei(M) + β

• ei>0

ei(M) M−

α,β(M) =

inf

αI≤A≤βI trAM = β

• ei<0

ei(M) + α

• ei>0

ei(M)

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Invariance with respect to reflection

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

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Invariance with respect to reflection

For a given unit vector e, let H(e) be the plane orthogonal to e and σ(e) the reflection with respect to e . F is invariant with respect 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 have the same eigenvalues. So any operator that depends only on the eigenvalues of the Hessian is invariant with respect to reflection and to rotation e.g. Pucci’s operators, the Laplace operator, Monge Ampere....

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Eigenfunctions of the Laplacian in the ball

∆u + λu = 0 in B , u = 0

• n ∂B ,

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

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Eigenfunctions of the Laplacian in the ball

On the other hand, the eigenfunctions corresponding to the second 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 the symmetries” of the domain and the operator.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Starting with Alexandrov and after the fundamental works of Serrin and Gidas, Ni, Nirenberg most results on symmetry of solutions rely on the moving plane method. It is impossible to even start mentioning all the results obtained via that method, be they for 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 .

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

• −∆u = f (x, u)

in Ω , u = 0

• n ∂Ω .

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

• −∆u = f (x, u)

in Ω , u = 0

• n ∂Ω .

Can the analogy be continued?

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

• −∆u = f (x, u)

in Ω , u = 0

• n ∂Ω .

Can the analogy be continued? when can one expect solutions of nonlinear equations to share the same symmetry of other eigenfunctions?

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

• −∆u = f (x, u)

in Ω , u = 0

• n ∂Ω .

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

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Limits of applications of the moving plane method.

• −∆u = f (x, u)

in Ω , u = 0

• n ∂Ω .

Sign changing solutions

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Limits of applications of the moving plane method.

• −∆u = f (x, u)

in Ω , u = 0

• n ∂Ω .

Sign changing solutions Non convex domains e.g. when Ω is an annulus.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Limits of applications of the moving plane method.

• −∆u = f (x, u)

in Ω , u = 0

• n ∂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 right monotonicity in the x variable.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Foliated Schwarz Symmetry

Definition Let B be a ball or an annulus in I

• Rn. A function u : B → I

R is foliated 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.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Lemma u is foliated Schwarz symmetric with respect to the direction p ∈ Sn−1 if and only if u(x) ≥ u(σe(x)) for all x ∈ B(e) and for every 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 and

• nly if there exists a direction e ∈ Sn−1 such that u is symmetric

with respect to H(e) and for any other direction e′ ∈ Sn−1 \ {±e}

• ne has either uθe,e′ ≥ 0 in B(e) or uθe,e′ ≤ 0 in B(e).

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

In the last decades, some work has been devoted to understanding under which conditions solutions of semilinear elliptic equations are foliated Schwarz symmetric. This line of research, which strongly relies on the maximum principle, was started by Pacella and then developed by Pacella and Weth. See also

• Gladiali, Pacella,Weth
• Pacella, Ramaswamy
• Weth .

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

• n ∂B .
• ne needs to study the sign of the first eigenvalue λ1(Lu, B(e)) of

the linearized operator Lu = ∆+ ∂f

∂u(|x|, u) at the solution u, in the

half domain B(e) = {x ∈ B : x · e > 0} for a direction e ∈ Sn−1.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

• n ∂B .
• ne needs to study the sign of the first eigenvalue λ1(Lu, B(e)) of

the linearized operator Lu = ∆+ ∂f

∂u(|x|, u) at the solution u, in the

half domain B(e) = {x ∈ B : x · e > 0} for a direction e ∈ Sn−1. Question: what plays the role of the linearized operator for the fully nonlinear problem?

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations 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

• n ∂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 .

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations 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

• n ∂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.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

The fully nonlinear operator Lu(v) : = M+

α,β(D2v) + ∂f

∂u (x, u) v . will be improperly called ”linearized operator at u”.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations 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 is radially symmetric and λ+

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

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations 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 is radially symmetric and λ+

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

Proposition Assume that Ω and F are symmetric with respect to the hyperplane H(e), let Ω(e) = Ω ∩ {x · e ≥ 0} (i) If f is convex and λ+

1 (Lu, Ω(±e)) > 0

• r

(ii) If f is strictly convex and λ+

1 (Lu, Ω(±e)) ≥ 0

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

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Definition of the eigenvalue.

Following the ideas of Berestycki, Nirenberg, Varadhan,

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Definition of the eigenvalue.

Following the ideas of Berestycki, Nirenberg, Varadhan, let λ+

1 (F, Ω) := sup{µ ∈ I

R, ∃ φ > 0 in Ω, F[φ] + µφ ≤ 0 }. λ−

1 (F, Ω) := sup{µ ∈ I

R, ∃ φ < 0 in Ω, F[φ] + µφ ≥ 0 }. with e.g. F[φ] = M+

α,β(D2φ) + c(x)φ.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

1 (F, Ω)φ = 0

in Ω φ = 0

• n ∂Ω

F[ψ] + λ−

1 (F, Ω)ψ = 0

in Ω ψ = 0

• n ∂Ω

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

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Proposition

With the above notations, the following properties hold: (i) If D1 ⊂ D2 and D1 = D2, then λ±

1 (D1) > λ± 1 (D2) .

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

k→+∞ 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 = λ±

1 is an eigenvalue then every corresponding

eigenfunction changes sign.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

Proposition

With the above notations, the following properties hold: (i) If D1 ⊂ D2 and D1 = D2, then λ±

1 (D1) > λ± 1 (D2) .

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

k→+∞ 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 = λ±

1 is an eigenvalue then every corresponding

eigenfunction 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 .

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction

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

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

Proposition Assume that Ω and F are symmetric with respect to the hyperplane H(e), (i) If f is convex and λ+

1 (Lu, Ω(±e)) > 0

• r

(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)

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Foliated Schwarz symmetry

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

• −F(x, D2u) = f (x, u)

in B , u = 0

• n ∂B ,

with f (x, ·) = f (|x|, ·) convex in I

• R. If there exists e ∈ Sn−1 such

that λ+

1 (Lu, B(e)) ≥ 0,

then u is foliated Schwarz symmetric.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations 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 Schwarz symmetric with respect to p ∈ Sn−1. Then, for all e ∈ Sn−1 such that e · p = 0, one has λ−

1 (Lu, B(e)) ≥ 0 .

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

1 (Lu, Ω(±e1)) ≥ 0

= ⇒ N(u) ∩ ∂Ω = ∅ . where N(u) is the nodal set of u.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Spectral properties

In any bounded domain Ω, one can define µ+

2 (Lu, Ω) = inf D⊂Ω max

• 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 second

eigenvalue of Lu. Corollary Under the above assumptions, if µ+

2 (Lu, Ω) ≥ 0 then u is foliated

Schwarz symmetric.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Spectral properties

In any bounded domain Ω, one can define µ+

2 (Lu, Ω) = inf D⊂Ω max

• 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 second

eigenvalue 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 two nodal regions. Proof: (On board)

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Nodal eigenvalues

This Proposition leads to believe that a natural candidate for being the second eigenvalue of Lu could be γ+

2 (Lu, B) = inf D⊂B max

• λ+

1 (Lu, D), λ− 1 (Lu, B \ D)

• ≥ µ+

2 (Lu, B) .

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Nodal eigenvalues

This Proposition leads to believe that a natural candidate for being the second eigenvalue of Lu could be γ+

2 (Lu, B) = inf D⊂B max

• λ+

1 (Lu, D), λ− 1 (Lu, B \ D)

• ≥ µ+

2 (Lu, B) .

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

2 (Lu, B) implies that u be foliated Schwarz

symmetric.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Nodal eigenvalues

For F uniformly elliptic and positively one homogeneous operator let Λ2(F) = inf{λ > max{λ−

1 (F), λ+ 1 (F)} : λ is an eigenvalue of F }.

(3) It was proved by Armstrong, that Λ2(F) > max{λ−

1 (F), λ+ 1 (F)}

and that for any µ ∈ (max{λ−

1 (F), λ+ 1 (F)}, Λ2(F))

and, for any continuous f , there exists a solution of the Dirichlet problem F(x, D2u) + µu = f (x) in B u = 0

• n ∂B.

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations 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 λr 2(F, B)

which denotes the smallest radial nodal eigenvalue of F in B. Theorem Let F be positively one homogeneous, then the following inequalities hold λr

2(F, B) > λ+ 1 (M+ α,β, B(e))

and Λ2(F, B) ≥ λ+

1 (M+ α,β, B(e)) .

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Nodal eigenvalues

To conclude, we observe that an important question which remains

• pen is whether λ+

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

B, 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

• n ∂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} = B(e).

Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Nodal eigenvalues