HB 2011

HB 2011

Fully nonlinear elliptic equations: Maximum principle, Hopf lemmaand qualitative properties of solutions.

I. Birindelli, F. Demengel

HB 2011

Introduction

In (1971) Serrin, introducing the moving plane method to PDEproved that if Ω is a bounded, C 2 domain such that there exists ua solution of

∆u = −1 in Ωu = 0 on ∂Ω∂u∂~n = c on ∂Ω,

then Ω is a ball and u is radial.

HB 2011

Introduction

Impossible to name all the extensions:

Exterior domains (Aftalion, Busca, Sirakov)

Quasilinear equations (Garofalo, Lewis, Farina,....)

Hyperbolic space (Kumaresan, Prajapat)

....

HB 2011

Introduction

Relationship with eigenvalue (El Soufi, Ilias; Pacard, Sicbaldi).”Extremal domains”, are defined as the critical points of the map

Ω −→ λ(Ω)

where λ(Ω) is the principal eigenvalue of −∆ with zero Dirichletboundary condition under the volume constraint |Ω| = cte.

A domain Ω is ”extremal” if and only if the eigenfunction φ > 0associated to λ(Ω) has constant Neumann boundary condition i.e.it is a solution of an overdetermined problem.

HB 2011

Introduction

Relationship with eigenvalue (El Soufi, Ilias; Pacard, Sicbaldi).”Extremal domains”, are defined as the critical points of the map

Ω −→ λ(Ω)

where λ(Ω) is the principal eigenvalue of −∆ with zero Dirichletboundary condition under the volume constraint |Ω| = cte.

A domain Ω is ”extremal” if and only if the eigenfunction φ > 0associated to λ(Ω) has constant Neumann boundary condition i.e.it is a solution of an overdetermined problem.

HB 2011

Introduction

A domain Ω is ”extremal” for the first eigenvalue functional underfixed volume variation if and only if the eigenfunction φ > 0associated to λ(Ω) has constant Neumann boundary condition i.e.it is a solution of an overdetermined problem.By Serrin’s result, this implies that bounded extremal domain areballs.

HB 2011

Introduction

Of course by Faber Krahn inequality

λ(B) ≤ λ(Ω) for |Ω| = |B|

the balls are not only critical values but they are minima.

HB 2011

Fullynonlinear

Following the idea of the acclaimed work of Berestycki, Nirenberg,Varadhan (1993), the concept of eigenvalue has been generalizedto fullynonlinear operators Armstrong, B., Busca, Demengel,Quaas, Ishii, Sirakov, Yoshimura and Lions.

HB 2011

Fullynonlinear

Joint work with F. Demengel

Hypothesis

Given a bounded domain Ω, given α > −1 we considerF (x ,∇u,D2u) satisfying:

(H1) F (x , tp, µX ) = |t|αµF (x , p,X ), ∀t ∈ IR?, µ ∈ IR+

(H2) a|p|αtrN ≤ F (x , p,M + N)− F (x , p,M) ≤ A|p|αtrN for0 < a ≤ A, α > −1 and N ≥ 0.

From now we will take

F (x ,∇u,D2u) := F [u] := |∇u|αM+a,A(D2u).

HB 2011

Fullynonlinear

Joint work with F. Demengel

Hypothesis

Given a bounded domain Ω, given α > −1 we considerF (x ,∇u,D2u) satisfying:

(H1) F (x , tp, µX ) = |t|αµF (x , p,X ), ∀t ∈ IR?, µ ∈ IR+

(H2) a|p|αtrN ≤ F (x , p,M + N)− F (x , p,M) ≤ A|p|αtrN for0 < a ≤ A, α > −1 and N ≥ 0.

From now we will take

F (x ,∇u,D2u) := F [u] := |∇u|αM+a,A(D2u).

HB 2011

Fullynonlinear

Joint work with F. Demengel

Hypothesis

Given a bounded domain Ω, given α > −1 we considerF (x ,∇u,D2u) satisfying:

(H1) F (x , tp, µX ) = |t|αµF (x , p,X ), ∀t ∈ IR?, µ ∈ IR+

(H2) a|p|αtrN ≤ F (x , p,M + N)− F (x , p,M) ≤ A|p|αtrN for0 < a ≤ A, α > −1 and N ≥ 0.

From now we will take

F (x ,∇u,D2u) := F [u] := |∇u|αM+a,A(D2u).

HB 2011

Fullynonlinear

Joint work with F. Demengel

Hypothesis

Given a bounded domain Ω, given α > −1 we considerF (x ,∇u,D2u) satisfying:

(H1) F (x , tp, µX ) = |t|αµF (x , p,X ), ∀t ∈ IR?, µ ∈ IR+

(H2) a|p|αtrN ≤ F (x , p,M + N)− F (x , p,M) ≤ A|p|αtrN for0 < a ≤ A, α > −1 and N ≥ 0.

From now we will take

F (x ,∇u,D2u) := F [u] := |∇u|αM+a,A(D2u).

HB 2011

Fullynonlinear

Joint work with F. Demengel

Hypothesis

Given a bounded domain Ω, given α > −1 we considerF (x ,∇u,D2u) satisfying:

(H1) F (x , tp, µX ) = |t|αµF (x , p,X ), ∀t ∈ IR?, µ ∈ IR+

(H2) a|p|αtrN ≤ F (x , p,M + N)− F (x , p,M) ≤ A|p|αtrN for0 < a ≤ A, α > −1 and N ≥ 0.

From now we will take

F (x ,∇u,D2u) := F [u] := |∇u|αM+a,A(D2u).

HB 2011

Fullynonlinear

F (x ,∇u,D2u) := F [u] := |∇u|αM+a,A(D2u).

Recall that the Pucci operators given by

M+a,A(M) = a

∑ei<0

ei + A∑ei>0

ei

whileM−

a,A(M) = a∑ei>0

ei + A∑ei<0

ei .

are

uniformly elliptic,

homogenous of degree 1,

invariant with respect to reflection with an hyperplane.

HB 2011

Fullynonlinear

Definition of the eigenvalue

Following the idea of Berestycki, Nirenberg, Varadhan, let

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

λ−(Ω) := supλ ∈ IR, ∃ φ < 0 in Ω, F [φ] + λ|φ|αφ ≥ 0 .

There exists φ > 0 (and ψ < 0) in Ω such that|∇φ|αM+

a,A(D2φ) + λ+(Ω)φα+1 = 0 in Ω

φ = 0 on ∂Ω

in the viscosity sense.

HB 2011

Fullynonlinear

Does the Faber Krahn inequality hold for fully-nonlinearoperators?

i.e. if |Ω| = |B| can we expect that

λ+(B) ≤ λ+(Ω)?

Or, more weakly, are balls critical points for the map

Ω −→ λ+(Ω)

under the volume constraint |Ω| = cte?First step, prove that Balls are the only bounded C 2 domains forwhich the eigenfunction satisfy both Dirichlet and Neumannboundary condition.

HB 2011

Fullynonlinear

(?)

|∇u|αM+

a,A(D2u) + f (u) = 0 in Ω

u = 0 on ∂Ω

Case 1 f is nonincreasing and C1, f (0) ≥ 0 .

Case 2 f (u) is a non decreasing function satisfying∀s > 1, ∀τ > 0, f (sτ) ≤ s1+αh(τ).

Case 3 α = 0, f is Lipschitz continuous.

Theorem (I. B.; F. Demengel) In these three cases, there exists aconstant δ which depends only on universal data and on f , suchthat for |a− A| < δ, if there exists u a constant sign C1 viscositysolution of the overdetermined problem (?), then

either c = f (0) = 0 ≡ u, or Ω is a ball, u is radial.

HB 2011

Fullynonlinear

(?)

|∇u|αM+

a,A(D2u) + f (u) = 0 in Ω

u = 0 on ∂Ω

Case 1 f is nonincreasing and C1, f (0) ≥ 0 .

Case 2 f (u) is a non decreasing function satisfying∀s > 1, ∀τ > 0, f (sτ) ≤ s1+αh(τ).

Case 3 α = 0, f is Lipschitz continuous.

Theorem (I. B.; F. Demengel) In these three cases, there exists aconstant δ which depends only on universal data and on f , suchthat for |a− A| < δ, if there exists u a constant sign C1 viscositysolution of the overdetermined problem (?), then

either c = f (0) = 0 ≡ u, or Ω is a ball, u is radial.

HB 2011

Fullynonlinear

(?)

|∇u|αM+

a,A(D2u) + f (u) = 0 in Ω

u = 0 on ∂Ω

Case 1 f is nonincreasing and C1, f (0) ≥ 0 .

Case 2 f (u) is a non decreasing function satisfying∀s > 1, ∀τ > 0, f (sτ) ≤ s1+αh(τ).

Case 3 α = 0, f is Lipschitz continuous.

Theorem (I. B.; F. Demengel) In these three cases, there exists aconstant δ which depends only on universal data and on f , suchthat for |a− A| < δ, if there exists u a constant sign C1 viscositysolution of the overdetermined problem (?), then

either c = f (0) = 0 ≡ u, or Ω is a ball, u is radial.

HB 2011

Fullynonlinear

Moving plane method relies on three facts

invariance with respect to reflection

comparison principle

Hopf’s lemma in corner

HB 2011

Fullynonlinear

Hopf’s lemma in corner

Suppose that f is C1 on IR+. Suppose that Ω? is some boundedC2,h domain, and suppose that H0 is an hyperplane such that thereexists P ∈ H0 ∩ ∂Ω?, with ~nΩ?(P) ∈ H0. Let Ω be the intersectionof Ω? with one of the half spaces bounded by H0.

Suppose that u and v are C2 solutions of|∇v |αMa,A(D2v) + f (v) ≤ |∇u|αMa,A(D2u) + f (u) in Ω,u < v in a neighborhood of P in Ω,u(P) = v(P) and either |∇u(P)| 6= 0 or |∇v(P)| 6= 0.

Then for any ~ν ∈ IRN ”entering” Ω

either ∂~νv(P) > ∂~νu(P) or ∂2~νv(P) > ∂2

~ν(P).

HB 2011

Fullynonlinear

Hopf’s lemma in corner

Suppose that f is C1 on IR+. Suppose that Ω? is some boundedC2,h domain, and suppose that H0 is an hyperplane such that thereexists P ∈ H0 ∩ ∂Ω?, with ~nΩ?(P) ∈ H0. Let Ω be the intersectionof Ω? with one of the half spaces bounded by H0.Suppose that u and v are C2 solutions of|∇v |αMa,A(D2v) + f (v) ≤ |∇u|αMa,A(D2u) + f (u) in Ω,u < v in a neighborhood of P in Ω,u(P) = v(P) and either |∇u(P)| 6= 0 or |∇v(P)| 6= 0.

Then for any ~ν ∈ IRN ”entering” Ω

either ∂~νv(P) > ∂~νu(P) or ∂2~νv(P) > ∂2

~ν(P).

HB 2011

Fullynonlinear

Hopf’s lemma in corner

Suppose that f is C1 on IR+. Suppose that Ω? is some boundedC2,h domain, and suppose that H0 is an hyperplane such that thereexists P ∈ H0 ∩ ∂Ω?, with ~nΩ?(P) ∈ H0. Let Ω be the intersectionof Ω? with one of the half spaces bounded by H0.Suppose that u and v are C2 solutions of|∇v |αMa,A(D2v) + f (v) ≤ |∇u|αMa,A(D2u) + f (u) in Ω,u < v in a neighborhood of P in Ω,u(P) = v(P) and either |∇u(P)| 6= 0 or |∇v(P)| 6= 0.

Then for any ~ν ∈ IRN ”entering” Ω

either ∂~νv(P) > ∂~νu(P) or ∂2~νv(P) > ∂2

~ν(P).

HB 2011

Fullynonlinear

(Sketch of the proof when α = 0).For any cone K , construct w > 0 such that w = 0 on ∂K , and inK

|∇w |αM−a,A(D2w) ≥ Cwα+1

so thatF [u + mw ]− Lf (u + mw) > F [v ]− Lf v

with v ≥ u + mw on the boundary of K . If ∂~νv(P) = ∂~νu(P) then

∂2~νv(P) = ∂2

~νu(P) + m∂2~νw(P).

This is useful if ∂2~νw(P) = 0 but this is not the case for the Pucci

operators.

HB 2011

Fullynonlinear

(Sketch of the proof when α = 0).For any cone K , construct w > 0 such that w = 0 on ∂K , and inK

|∇w |αM−a,A(D2w) ≥ Cwα+1

so thatF [u + mw ]− Lf (u + mw) > F [v ]− Lf v

with v ≥ u + mw on the boundary of K . If ∂~νv(P) = ∂~νu(P) then

∂2~νv(P) = ∂2

~νu(P) + m∂2~νw(P).

This is useful if ∂2~νw(P) = 0 but this is not the case for the Pucci

operators.

HB 2011

Fullynonlinear

(Sketch of the proof when α = 0).For any cone K , construct w > 0 such that w = 0 on ∂K , and inK

|∇w |αM−a,A(D2w) ≥ Cwα+1

so thatF [u + mw ]− Lf (u + mw) > F [v ]− Lf v

with v ≥ u + mw on the boundary of K . If ∂~νv(P) = ∂~νu(P) then

∂2~νv(P) = ∂2

~νu(P) + m∂2~νw(P).

This is useful if ∂2~νw(P) = 0 but this is not the case for the Pucci

operators.

HB 2011

Fullynonlinear

Let w = rγψ(θ), where |x | = r and θ are coordinates on thesphere:

D2w = rγ−2(ψθiθj r

2(∇θi ⊗∇θj) +

+γψθi (x ⊗∇θi +∇θi ⊗ x)

+ψθi r2(D2θi ) +

+γψ(I + (γ − 2)

x

r⊗ x

r

)),

HB 2011

Fullynonlinear

Let w = rγψ(θ), where |x | = r and θ are coordinates on thesphere:

M−a,A(D2w) ≥ rγ−2(H(θ,∇ψ,D2ψ) + aγ(N + γ − 2)ψ)

Here H is an elliptic fullynonlinear operator.

HB 2011

Fullynonlinear

M−a,A(D2w) ≥ rγ−2(H(θ,∇ψ,D2ψ) + aγ(N + γ − 2)ψ)

For H we can define the principal eigenvalue as above. So that for

aγ(γ + N − 2) = λ+

w is the right sub-solution.

The problem is that γ > 2. Observe

that for w(x) = x1x2, D2w =

(0 11 0

)so that

0 > a− A =M−a,A(D2w)

≥ 2γ−2(H(θ,∇ψ,D2ψ) + a2Nψ)

by definition of λ+, λ+ > 2aN which implies that γ > 2.

HB 2011

Fullynonlinear

M−a,A(D2w) ≥ rγ−2(H(θ,∇ψ,D2ψ) + aγ(N + γ − 2)ψ)

For H we can define the principal eigenvalue as above. So that for

aγ(γ + N − 2) = λ+

w is the right sub-solution. The problem is that γ > 2.

Observe

that for w(x) = x1x2, D2w =

(0 11 0

)so that

0 > a− A =M−a,A(D2w)

≥ 2γ−2(H(θ,∇ψ,D2ψ) + a2Nψ)

by definition of λ+, λ+ > 2aN which implies that γ > 2.

HB 2011

Fullynonlinear

M−a,A(D2w) ≥ rγ−2(H(θ,∇ψ,D2ψ) + aγ(N + γ − 2)ψ)

For H we can define the principal eigenvalue as above. So that for

aγ(γ + N − 2) = λ+

w is the right sub-solution. The problem is that γ > 2. Observe

that for w(x) = x1x2, D2w =

(0 11 0

)so that

0 > a− A =M−a,A(D2w)

≥ 2γ−2(H(θ,∇ψ,D2ψ) + a2Nψ)

by definition of λ+, λ+ > 2aN which implies that γ > 2.

HB 2011

Fullynonlinear

So, in order to conclude, we shall use C 2,β regularity of thesolution. Indeed suppose by contradiction that

∂2~νv(P) = ∂2

~νu(P)

we get that

c |P − Q|γ−2 ≤ ∂2~νv(Q)− ∂2

~νv(P) ≤ C |P − Q|β.

This is a contradiction if γ − 2 < β.

This is why we require A− asmall enough.

HB 2011

Fullynonlinear

So, in order to conclude, we shall use C 2,β regularity of thesolution. Indeed suppose by contradiction that

∂2~νv(P) = ∂2

~νu(P)

we get that

c |P − Q|γ−2 ≤ ∂2~νv(Q)− ∂2

~νv(P) ≤ C |P − Q|β.

This is a contradiction if γ − 2 < β. This is why we require A− asmall enough.

HB 2011

Fullynonlinear

In a recent preprint of Armstrong, Smart, Sirakov, (arxive 2011)there is a construction of barrier functions in cones which isdifferent from our w , and can be used instead to compute γ.

HB 2011

Fullynonlinear

Direction of research

Prove that extremal domains are domains for which theeigenfunction satisfies an overdetermined problem

Recently, Helein, Hauswirth and Pacard, have constructed indimension 2 an exeptional domain which is not radial (i.e.such that there exists a Harmonic function which satisfiesboth O Dirichlet and Neumann condition on the boundary) .Are there such exeptional domain for the Pucci operators.

HB 2011

Fullynonlinear

Direction of research

Prove that extremal domains are domains for which theeigenfunction satisfies an overdetermined problem

Recently, Helein, Hauswirth and Pacard, have constructed indimension 2 an exeptional domain which is not radial (i.e.such that there exists a Harmonic function which satisfiesboth O Dirichlet and Neumann condition on the boundary) .Are there such exeptional domain for the Pucci operators.