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 of 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 operator are inherited by the viscosity solutions of the homogeneous Dirichlet problem � − F ( x , D 2 u ) = f ( x , u ) in Ω , (1) u = 0 on ∂ Ω , R n , n ≥ 2, is a bounded domain and F is a fully where Ω ⊂ I 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 ∈ S n , 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 tr AM = α e i ( M ) + β e i ( M ) α I ≤ A ≤ β I e i < 0 e i > 0 M − � � α,β ( M ) = α I ≤ A ≤ β I tr AM = β inf e i ( M ) + α e i ( M ) e i < 0 e i > 0
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 ( σ e ( x ) , ( I − 2 e ⊗ e ) M ( I − 2 e ⊗ e )) = F ( x , M ) ∀ x ∈ Ω , M ∈ S n F is invariant with respect to rotation if � Ox , OMO T � F = F ( x , M ) ∀ x ∈ Ω , M ∈ S n all orthogonal matrix O
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 ( σ e ( x ) , ( I − 2 e ⊗ e ) M ( I − 2 e ⊗ e )) = F ( x , M ) ∀ x ∈ Ω , M ∈ S n F is invariant with respect to rotation if � Ox , OMO T � F = F ( x , M ) ∀ x ∈ Ω , M ∈ S n all orthogonal matrix O Observe that M and ( I n − 2 e ⊗ e ) M ( I n − 2 e ⊗ e ) and OMO T 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 + λ u = 0 in B , u = 0 on ∂ 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 φ 2 ( 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 on ∂ 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 on ∂ 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 on ∂ 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 on ∂ 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 on ∂ 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 on ∂ 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 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 right monotonicity in the x variable.
Symmetry and spectral properties for viscosity solutions of fully nonlinear equations Introduction Foliated Schwarz Symmetry Definition R n . A function u : B → I Let B be a ball or an annulus in I R is foliated Schwarz symmetric if there exists a unit vector p ∈ S n − 1 � � x such that u ( x ) only depends on | x | and θ = arccos | 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 ∈ S n − 1 if and only if u ( x ) ≥ u ( σ e ( x )) for all x ∈ B ( e ) and for every e ∈ S n − 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 ∈ S n − 1 if and only if u ( x ) ≥ u ( σ e ( x )) for all x ∈ B ( e ) and for every e ∈ S n − 1 such that e · p ≥ 0 . Whose consequence is Proposition A function u ∈ C 1 ( B ) ∩ C ( B ) is foliated Schwarz symmetric if and only if there exists a direction e ∈ S n − 1 such that u is symmetric with respect to H ( e ) and for any other direction e ′ ∈ S n − 1 \ {± e } one has either u θ e , e ′ ≥ 0 in B ( e ) or u θ e , 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 on ∂ B . one needs to study the sign of the first eigenvalue λ 1 ( L u , B ( e )) of the linearized operator L u = ∆+ ∂ f ∂ u ( | x | , u ) at the solution u , in the half domain B ( e ) = { x ∈ B : x · e > 0 } for a direction e ∈ S n − 1 .
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