Wentzell eigenvalues, upper bound and stability

Wentzell eigenvalues, upper bound and stability

Jimmy LAMBOLEY University Paris-Dauphine

with M. Dambrine, D. Kateb

Isoperimetric Problems Between Analysis and Geometry

16/06/2014, Pisa

Wentzell eigenvalue problem

−∆u = 0 in Ω

−β∆

u + ∂

u = λu on ∂Ω where β ∈ R

.

It has a discrete sprectrum consisting in a sequence

λ

(Ω) = 0 < λ

(Ω) ≤ λ

(Ω) . . . → +∞ Remark : it is the spectrum of

−β∆

+ D

where D is the Dirichlet-to-Neumann operator on ∂Ω.

Question : Does the ball maximize λ

among smooth open sets of

given volume ?

Wentzell eigenvalue problem

−∆u = 0 in Ω

−β∆

u + ∂

u = λu on ∂Ω where β ∈ R

.

It has a discrete sprectrum consisting in a sequence

λ

(Ω) = 0 < λ

(Ω) ≤ λ

(Ω) . . . → +∞

Remark : it is the spectrum of

−β∆

+ D

where D is the Dirichlet-to-Neumann operator on ∂Ω.

Question : Does the ball maximize λ

among smooth open sets of

given volume ?

Wentzell eigenvalue problem

−∆u = 0 in Ω

−β∆

u + ∂

u = λu on ∂Ω where β ∈ R

.

It has a discrete sprectrum consisting in a sequence

λ

(Ω) = 0 < λ

(Ω) ≤ λ

(Ω) . . . → +∞

Remark : it is the spectrum of

−β∆

+ D

where D is the Dirichlet-to-Neumann operator on ∂Ω.

Question : Does the ball maximize λ

among smooth open sets of

given volume ?

Wentzell eigenvalue problem

−∆u = 0 in Ω

−β∆

u + ∂

u = λu on ∂Ω where β ∈ R

.

It has a discrete sprectrum consisting in a sequence

λ

(Ω) = 0 < λ

(Ω) ≤ λ

(Ω) . . . → +∞

Remark : it is the spectrum of

−β∆

+ D

where D is the Dirichlet-to-Neumann operator on ∂Ω.

Variational formulation

A

(u, v) = Z

Ω

∇u.∇v dx + β Z

∂Ω

∇

u.∇

v d σ, B (u, v ) = Z

∂Ω

uv ,

λ

(Ω) = min

A

(v, v )

B(v , v) , v ∈ H

(Ω), Z

∂Ω

v = 0

Outline

1

Extremal cases β = 0 and β = ∞

2

Generalization of Brock’s bound

3

First order analysis

4

Second order analysis

Extremal casesβ= 0andβ=

Outline

1

Extremal cases β = 0 and β = ∞

2

Generalization of Brock’s bound

3

First order analysis

4

Second order analysis

Extremal casesβ= 0andβ=∞

Steklov eigenvalue problem, β = 0

∆u = 0 in Ω

∂

u = λ

u on ∂Ω It has a discrete sprectrum consisting in a sequence

λ

(Ω) = 0 < λ

(Ω) ≤ λ

(Ω) . . . → +∞

Remark :

λ

(αΩ) = α

λ

(Ω).

Extremal casesβ= 0andβ=

Bound for λ ^St ₁ , β = 0, Dimension 2

Weinstock (1954) ; Ω simply connected λ

(Ω) ≤ 2π

P (Ω)

≤ r π

|Ω|

,

Hersch-Payne (1968) ; Ω simply connected 1

λ

(Ω) + 1

λ

(Ω) ≥ P (Ω) π , Hersch-Payne-Schiffer (1975) ; Ω simply connected

λ

(Ω).λ

(Ω) ≤ 4π

P (Ω)

,

Extremal casesβ= 0andβ=∞

Bound for λ ^St ₁ , β = 0, Dimension d

Brock (2001) : Ω any smooth set in R

such that R

∂Ω

x = 0 :

d

X

i=1

1

λ

(Ω) ≥ 1

|Ω|

Z

∂Ω

|x|

≥ d |Ω|

ω

.

λ

(Ω) ≤ d |Ω|

R

∂Ω

|x|

≤ ω

|Ω|

.

Extremal casesβ= 0andβ=

Laplace-Beltrami eigenvalue problem, β = +∞

−∆

u = λu on ∂Ω

λ

(∂Ω) = 0 < λ

(∂Ω) ≤ λ

(∂Ω) . . . → +∞

Remarks :

λ

(α∂Ω) = α

λ

(∂Ω).

λ

(Ω) = lim

β

.

Extremal casesβ= 0andβ=∞

Bound for λ ^LB ₁ , β = +∞, 2-dimensional surface

Hersch (1970) : If Ω ⊂ R

smooth and bounded is such that ∂Ω is diffeomorphic to the 2-dimensional sphere S

= ∂B, then

1

λ

(∂Ω) + 1

λ

(∂Ω) + 1

λ

(∂Ω) ≥ 3 8π P (Ω).

λ

(∂Ω)P (Ω) ≤ λ

( S

)P (B ).

Colbois-Dryden-El Soufi (2009) sup

Ω⊂R³

λ

(∂Ω)P(Ω) = ∞

where the supremum is taken among smooth compact set Ω.

Extremal casesβ= 0andβ=

Bound for λ ^LB ₁ , β = +∞, 2-dimensional surface

Hersch (1970) : If Ω ⊂ R

smooth and bounded is such that ∂Ω is diffeomorphic to the 2-dimensional sphere S

= ∂B, then

1

λ

(∂Ω) + 1

λ

(∂Ω) + 1

λ

(∂Ω) ≥ 3 8π P (Ω).

λ

(∂Ω)P (Ω) ≤ λ

( S

)P (B ).

Colbois-Dryden-El Soufi (2009) sup

Ω⊂R³

λ

(∂Ω)P(Ω) = ∞

where the supremum is taken among smooth compact set Ω.

Extremal casesβ= 0andβ=∞

Bound for λ ^LB ₁ , β = +∞, 2-dimensional surface, volume constraint

Corollary

If Ω ⊂ R

smooth and bounded is such that ∂Ω is diffeomorphic to the 2-dimensional sphere S

= ∂B, then

λ

(∂Ω)|Ω|

≤ λ

(S

)|B |

.

Extremal casesβ= 0andβ=∞

Bound for λ ^LB ₁ , β = +∞, m-dimensional manifold

Colbois-Dodziuk (1994) ; for m ≥ 3, sup

M

λ

(M)Vol(M )

= ∞

where the supremum is taken among smooth compact manifold of dimension m and fixed smooth structure.

Colbois-Dryden-El Soufi (2009) ; for m ≥ 3, sup

Ω⊂R^m+1

λ

(∂Ω)P (Ω)

= ∞

where the supremum is taken among smooth compact set Ω such

that ∂Ω has a fixed smooth structure.

Extremal casesβ= 0andβ=∞

Bound for λ ^LB ₁ , β = +∞, m-dimensional manifold

Colbois-Dodziuk (1994) ; for m ≥ 3, sup

M

λ

(M)Vol(M )

= ∞

where the supremum is taken among smooth compact manifold of dimension m and fixed smooth structure.

Colbois-Dryden-El Soufi (2009) ; for m ≥ 3, sup

Ω⊂R^m+1

λ

(∂Ω)P (Ω)

= ∞

Outline

1

Extremal cases β = 0 and β = ∞

2

Generalization of Brock’s bound

3

First order analysis

4

Second order analysis

Generalization of Brock’s bound

Generalization of Brock’s result

Theorem (Dambrine-Kateb-L. 2014) Let Ω a smooth set such that R

∂Ω

x = 0. Let Λ[Ω] be the spectral radius of P [Ω] = (p

)

defined as

p

= Z

∂Ω

(δ

− n

n

),

where n is the outward normal vector to ∂Ω. Then if β ≥ 0, one has :

d

X

i=1

1 λ

(Ω) ≥

R

∂Ω

|x|

|Ω| + βΛ[Ω] (1)

Generalization of Brock’s result

Corollary

If β ≥ 0, it holds :

λ

(Ω) ≤ d |Ω| + βΛ[Ω]

R

∂Ω

|x|

Equality holds if Ω is a ball.

β = 0 is Brock’s result.

In general, the right-hand side is not maximized by the ball.

β = ∞ gives :

λ

(∂Ω) ≤ d Λ[Ω]

R

∂Ω

|x|

Generalization of Brock’s bound

Proof of the generalization of Brock’s result

Variational formulation

A

(u, v) = Z

Ω

∇u.∇v dx + β Z

∂Ω

∇

u.∇

v d σ, B (u, v ) = Z

∂Ω

uv ,

d

X

i=1

1

λ

(Ω) = max

v1,···,v_d d

X

i=1

B(v

, v

)

A

(v

, v

) ,

where the functions (v

)

are non zero functions that are

B -orthogonal to the constants and pairwise A

-orthogonal.

Proof of the generalization of Brock’s result

Choice of test functions

We choose coordinates such that

∀i 6= j , Z

∂Ω

x

= 0 and Z

∂Ω

x

x

= 0.

Let for i ∈ J 1, d K : w

=

d

X

j=1

c

x

, B-orthogonal to the constants.

Generalization of Brock’s bound

Proof of the generalization of Brock’s result

Choice of test functions

Let us compute A

(w

, w

) : Z

∂Ω

∇

w

· ∇

w

= X

k,m

c

p

c

= (CP [Ω]C

)

. Therefore

A

(w

, w

) = |Ω| (CC

)

+ β(CP[Ω]C

)

We choose C ∈ O(n) such that CP[Ω]C

is diagonal. Then

Proof of the generalization of Brock’s result

Conclusion

Using

B(w

, w

) =

d

X

k=1

c

Z

∂Ω

x

we obtain :

d

X

i=1

1 λ

(Ω) ≥

d

X

k=1

Z

∂Ω

x

d

X

i=1

c

!

|Ω| + βΛ[Ω] = Z

∂Ω

|x|

|Ω| + βΛ[Ω]

First order analysis

Outline

1

Extremal cases β = 0 and β = ∞

2

Generalization of Brock’s bound

3

First order analysis

4

Second order analysis

Derivative of a multiple eigenvalue

Theorem (Dambrine-Kateb-L. 2014)

Let λ be an eigenvalue of order m and Ω

= (I + t V)(Ω). Then there exist m functions (t 7→ λ

(t))

such that

λ

(0) = λ,

for |t | small, λ

(t) is an eigenvalue of Ω

, the functions (t 7→ λ

(t))

J1,mK

admit derivatives and their values at 0 are the eigenvalues of the matrix M= M

(V

) :

M

= Z

∂Ω

V

∇

u

.∇

u

− ∂

u

∂

u

− λHu

u

+ β HI

− 2D

b

∇

u

.∇

u

d σ.

where (u

)

denote the eigenfunctions associated to λ.

First order analysis

First order optimality for the ball

If Ω = B the unit ball and V preserves the volume, then : M

= C (d , β)

Z

S^d−1

V

x

x

.

Corollary

Any ball B is a critical shape for λ

with volume constraint : for every volume preserving deformations V,

d

X

k=1

λ

(0) = Tr(M

(V

)) = 0.

Outline

1

Extremal cases β = 0 and β = ∞

2

Generalization of Brock’s bound

3

First order analysis

4

Second order analysis

Second order analysis

Sign of the second order derivative

Theorem (Dambrine-Kateb-L. 2014)

Let B be a ball in R

or R

and t 7→ B

= I + tV + O (t

) a volume preserving deformation such that

V

is orthogonal to spherical harmonics of order 2.

Then the functions (t 7→ λ

(t))

J1,dK

admit a second derivative and there exists α > 0 such that

d

X λ

(0) ≤ −αkV

k

.

Local optimality

Corollary

If B is a ball in R

or R

, and t 7→ B

= (I + tV + O(t

))(B) a smooth volume preserving deformation, then

λ

(B) ≥ λ

(B

), for t small enough.

Second order analysis

Perspectives - Open Problems

Positive answer to the question when Ω ⊂ R

and ∂Ω is differomorphic to S

?

Study the stability question for Hersch’s inequality in a smooth neighborhood :

- solve the “two-norm discrepancy issue” :

S (Ω) :=

d

X

i=1

1

λ

(Ω) = S (B) + S

(B)(V , V )

| {z }

≤−αkVnk²

H1

+o (kV k

).

- prove coercivity for deformation preserving the perimeter.