Alexandre Girouard

From Steklov to Neumann via homogenization

NeumannConcentration of density // Steklov

Homogenizationtt

Dynamical

β→∞

jj

��Applications to isoperimetric problems

**tt ��Planar domains Higher dimensions Riemannian manifolds

joint work with

Antoine Henrot (Institut Elie Cartan) and Jean Lagacé (University College London)

1. Neumann, Steklov and dynamical problems

Let Ω ⊂ Rd be open, bounded with smooth ∂Ω

Neumann{−∆u = µu in Ω∂νu = 0 on ∂Ω

0 = µ0 ≤ µ1 ≤ µ2 ≤ · · · ↗ +∞

Steklov{∆u = 0 in Ω

∂νu = σu on ∂Ω

0 = σ0 ≤ σ1 ≤ σ2 ≤ · · · ↗ +∞

Dynamical with parameter β ≥ 0{−∆U = βΣU in Ω∂νU = ΣU on ∂Ω

0 = Σ0,β ≤ Σ1,β ≤ Σ2,β ≤ · · · ↗ ∞

Reference: François – von Below (2005)

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1. Neumann, Steklov and dynamical problems

Let Ω ⊂ Rd be open, bounded with smooth ∂Ω

Neumann{−∆u = µu in Ω∂νu = 0 on ∂Ω

0 = µ0 ≤ µ1 ≤ µ2 ≤ · · · ↗ +∞

Steklov{∆u = 0 in Ω

∂νu = σu on ∂Ω

0 = σ0 ≤ σ1 ≤ σ2 ≤ · · · ↗ +∞

Dynamical with parameter β ≥ 0{−∆U = βΣU in Ω∂νU = ΣU on ∂Ω

0 = Σ0,β ≤ Σ1,β ≤ Σ2,β ≤ · · · ↗ ∞

Reference: François – von Below (2005)

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Mass concentration along ∂Ω

Let w : Ω −→ R be a positive density

Consider the non-homogeneous Neumann problem{−∆u = µwu in Ω∂νu = 0 on ∂Ω

If the density w concentrates on ∂Ω, then µk(Ω,w)→ σk(Ω)

References: Bandle 1980 Arrieta – Jiménez-Casas – Rodríguez-Bernal 2008 Lamberti – Provenzano 2014

3 / 23

Neumannwdx

?−⇀ dH∂Ω // Steklov

wwDynamical

gg

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2. Homogenization theory

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Let Ω ⊂ Rd be open, bounded with smooth ∂Ω

Given ε > 0 and k ∈ Zd

Bεk = εk+[− ε

2,ε

2

]dTεk = B(εk, rε)

Iε = {k ∈ Zd : Bεk ⊂ Ω}

Ωε = Ω \⋃k∈Iε

Tεk

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Ωε = Ω \ Tε where Tε =⋃k∈Iε

B(εk, rε)

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Girouard – Henrot – Lagacé2019

Ωε = Ω \ Tε where Tε = ∪k∈IεB(εk, rε)

Ad := |∂B(0,1)|

Dynamical problem with parameter β ≥ 0{−∆U = βΣU in Ω∂νU = ΣU on ∂Ω

Main homogenization theorem

rd−1ε ∼β

Adεd =⇒

|∂Tε| ε→0−−−→ β|Ω|

σk(Ωε)

ε→0−−−→ Σk,β

Reference: arXiv:1906.09638

8 / 23

Girouard – Henrot – Lagacé2019

Ωε = Ω \ Tε where Tε = ∪k∈IεB(εk, rε)

Ad := |∂B(0,1)|

Dynamical problem with parameter β ≥ 0{−∆U = βΣU in Ω∂νU = ΣU on ∂Ω

Main homogenization theorem

rd−1ε ∼β

Adεd =⇒

|∂Tε| ε→0−−−→ β|Ω|

σk(Ωε)

ε→0−−−→ Σk,β

Reference: arXiv:1906.09638

8 / 23

TrichotomyAnalogous to the crushed ice problem

Ωε = Ω \ Tε where Tε =⋃k∈Iε

B(εk, rε)

Critical regime

If rd−1ε ∼ A−1d βεd, then |∂Tε| → β|Ω| and σk(Ωε)→ Σk,β

Small holes, β = 0Thawing

If rd−1ε = o(εd), then |∂Tε| → 0 and σk(Ωε)→ σk(Ω)

Large holesFreezing

If limε→0

rd−1ε /εd = +∞, then |∂Tε| → ∞ and σk(Ωε)→ 0

9 / 23

Girouard – Henrot – Lagacé2019

Neumannwdx

?−⇀ dH∂Ω // Steklov

homogenizationwwDynamical

gg

Theorem

limβ→∞

βΣk,β = µk

Reference: arXiv:1906.09638

10 / 23

Girouard – Henrot – Lagacé2019

Neumannwdx

?−⇀ dH∂Ω // Steklov

homogenizationwwDynamical

β→∞

gg

Theorem

limβ→∞

βΣk,β = µk

Reference: arXiv:1906.09638

10 / 23

Corollary for planar domains

A2 = |∂D| = 2π rε ∼β

2πε2

|∂Ωε| = |∂Ω|+ |∂Tε| ε→0−−−→ |∂Ω|+ β|Ω|

Area-Normalized Neumann

σk(Ωε)|∂Ωε| ε→0−−−→ Σk,β(|∂Ω|+ β|Ω|)

β→∞−−−−→ µk(Ω)|Ω|

Perimeter-Normalized Steklov

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Corollary for planar domains

A2 = |∂D| = 2π rε ∼β

2πε2

|∂Ωε| = |∂Ω|+ |∂Tε| ε→0−−−→ |∂Ω|+ β|Ω|

Area-Normalized Neumann

σk(Ωε)|∂Ωε| ε→0−−−→ Σk,β(|∂Ω|+ β|Ω|)

β→∞−−−−→ µk(Ω)|Ω|

Perimeter-Normalized Steklov

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3. Applications to isoperimetric problems

If Ω ⊂ R2 is simply-connected then. . .

Szegő (1954)

µ1(Ω)|Ω| ≤ µ1(D)× π ∼= 3.39π

Weinstock (1954)

σ1(Ω)|∂Ω| ≤ σ1(D)× 2π = 2π

with equality if and only if Ω is a disk

If Ω ⊂ R2 is simply-connected then. . .

Weinberger (1956)

µ1(Ω)|Ω| ≤ µ1(D)× π

Kokarev (2014)

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3. Applications to isoperimetric problems

If Ω ⊂ R2 is simply-connected then. . .

Szegő (1954)

µ1(Ω)|Ω| ≤ µ1(D)× π ∼= 3.39π

Weinstock (1954)

σ1(Ω)|∂Ω| ≤ σ1(D)× 2π = 2π

with equality if and only if Ω is a disk

If Ω ⊂ R2 is simply-connected then. . .

Weinberger (1956)

µ1(Ω)|Ω| ≤ µ1(D)× π

Kokarev (2014)

σ1(Ω)|∂Ω| < 8π

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3. Applications to isoperimetric problems

If Ω ⊂ R2 is simply-connected then. . .

Szegő (1954)

µ1(Ω)|Ω| ≤ µ1(D)× π ∼= 3.39π

Weinstock (1954)

σ1(Ω)|∂Ω| ≤ σ1(D)× 2π = 2π

with equality if and only if Ω is a disk

If Ω ⊂ R2 is simply-connected then. . .

Weinberger (1956)

µ1(Ω)|Ω| ≤ µ1(D)× π

Kokarev (2014)

?? σ1(Ω)|∂Ω| < 2π ??

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Let Ωε := B(0,1) \ B(0, ε)

Dittmar (2004)σ1(Ωε) = σ1(D) + o(ε) = 1 + o(ε)

This implies σ1(Ωε)|∂Ωε| = 2π + 2πε+ o(ε) > 2π

For ε ∼= 0.2, we have σ1(Ωε)|∂Ωε| ∼= 2.17π13 / 23

How large can σ1(Ω)|∂Ω| be for Ω ⊂ R2 ?

2.17π < sup{σ1(Ω)|∂Ω| : Ω ⊂ R2} ≤ 8π

Homogenization of Ω = D

Perimeter-Normalized Steklov

σ1(Ωε)|∂Ωε| ε→0−→ Σ1,β(|∂Ω|+ β|Ω|)

β→∞−→ µ1(Ω)|Ω| = µ1(D)|D|

Area-Normalized Neumann

Corollary

3.39π ≈ µ1(D)π ≤ sup{σ1(Ω)|∂Ω| : Ω ⊂ R2} ≤ 8π

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How large can σ1(Ω)|∂Ω| be for Ω ⊂ R2 ?

2.17π < sup{σ1(Ω)|∂Ω| : Ω ⊂ R2} ≤ 8π

Homogenization of Ω = D

Perimeter-Normalized Steklov

σ1(Ωε)|∂Ωε| ε→0−→ Σ1,β(|∂Ω|+ β|Ω|)

β→∞−→ µ1(Ω)|Ω| = µ1(D)|D|

Area-Normalized Neumann

Corollary

3.39π ≈ µ1(D)π ≤ sup{σ1(Ω)|∂Ω| : Ω ⊂ R2} ≤ 8π

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How large can σk(Ω)|∂Ω| be for Ω ⊂ R2 ?

Conjecture 1

There is no isoperimetric maximizer for σ1

Conjecture 2

sup{σ1(Ω)|∂Ω|} = µ1(D)× π ∼= 3.39π

Conjecture 3

sup{σk(Ω)|∂Ω|} = sup{µk(Ω)|Ω|}

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How large can σk(Ω)|∂Ω| be for Ω ⊂ R2 ?

Conjecture 1

There is no isoperimetric maximizer for σ1

Conjecture 2

sup{σ1(Ω)|∂Ω|} = µ1(D)× π ∼= 3.39π

Conjecture 3

sup{σk(Ω)|∂Ω|} = sup{µk(Ω)|Ω|}

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How large can σk(Ω)|∂Ω| be for Ω ⊂ R2 ?

Conjecture 1

There is no isoperimetric maximizer for σ1

Conjecture 2

sup{σ1(Ω)|∂Ω|} = µ1(D)× π ∼= 3.39π

Conjecture 3

sup{σk(Ω)|∂Ω|} = sup{µk(Ω)|Ω|}

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Isoperimetric bounds: Steklov −→ Neumann

Corollary

Let Ω ⊂ R2 be open, bounded with smooth ∂ΩThen µ1(Ω)|Ω| ≤ 8π

Proof.It follows from the above that Kokarev

µ1(Ω)|Ω| = limβ→∞

limε→0

< 8π︷ ︸︸ ︷σ1(Ω

ε)|∂Ωε|

This inequality is of course not new

The moral is newBounds on σk which depends only on the perimeter can be

used to prove bounds on µk

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Isoperimetry in higher dimensions: Ω ⊂ Rd

Colbois – El Soufi – Girouard (2011)There exists a constant C = C(d) such that

σk(Ω)|∂Ω| ≤ C|Ω|1−2d k2/d (1)

Applying to Ωε ⊂ Ω and taking ε→ 0 gives

Σk,β(|∂Ω|+ β|Ω|) ≤ C|Ω|1−2d k2/d

CorollaryKröger (1992)

µk(Ω)|Ω|2/d ≤ Ck2/d

Corollary2/d is the optimal exponent in (1)

Any smaller exponent would contradict Weyl’s law for µk

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Isoperimetry in higher dimensions: Ω ⊂ Rd

Colbois – El Soufi – Girouard (2011)There exists a constant C = C(d) such that

σk(Ω)|∂Ω| ≤ C|Ω|1−2d k2/d (1)

Applying to Ωε ⊂ Ω and taking ε→ 0 gives

Σk,β(|∂Ω|+ β|Ω|) ≤ C|Ω|1−2d k2/d

CorollaryKröger (1992)

µk(Ω)|Ω|2/d ≤ Ck2/d

Corollary2/d is the optimal exponent in (1)

Any smaller exponent would contradict Weyl’s law for µk

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4. Homogenization on a closed Riemannian manifold M

Joint work with Jean Lagacé (University College London)

M : closed Riemannian manifold, β ∈ C∞(M) positive

−∆f = βλf

0 = λ0 ≤ λ1(M, β) ≤ λ2(M, β) ≤ · · · ↗ ∞.

TheoremThere is a sequence Ωε ⊂ M of domains such that

dH∂Ωε?−⇀ β dx and σk(Ωε)

ε→0−−−⇀ λk(M, β)

CorollaryIf M is a surface, there is Ωε ⊂ M such that

σk(Ωε)|∂Ωε| ε→0−−−⇀ λk(M)A(M).

Reference: arXiv: coming next week. . .

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5. Proofs

Variational characterizations

Eigenfunctions are the critical points of the Dirichlet energy:

H1(Ω) 3 u 7−→∫Ω|∇u|2

Under the following constraints

Dynamical∫Ωu2 + 2πβ

∫∂Ω

u2 = 1.

Steklov∫∂Ω

u2 = 1.

(U,Σ) is a dynamical eigenpair

⇐⇒∫Ω

∇U·∇v = Σ(2πβ∫Ω

Uv+

∫∂Ω

Uv)

(u, σ) is a Steklov eigenpair

⇐⇒∫Ω

∇u · ∇v = σ∫∂Ω

uv

∀v ∈ C∞(Ω)

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5. Proofs

Variational characterizations

Eigenfunctions are the critical points of the Dirichlet energy:

H1(Ω) 3 u 7−→∫Ω|∇u|2

Under the following constraints

Dynamical∫Ωu2 + 2πβ

∫∂Ω

u2 = 1.

Steklov∫∂Ω

u2 = 1.

(U,Σ) is a dynamical eigenpair

⇐⇒∫Ω

∇U·∇v = Σ(2πβ∫Ω

Uv+

∫∂Ω

Uv)

(u, σ) is a Steklov eigenpair

⇐⇒∫Ω

∇u · ∇v = σ∫∂Ω

uv

∀v ∈ C∞(Ω)

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StrategySuppose rε ∼ βε2.

For ε > 0, let uεk be a complete set of Stekov eigenfunctions

{∆uεk = 0 in Ω

ε

∂νuεk = σk

εuεk on ∂Ωε

∫∂Ωε|uk|2 = 1.

Define Uεk ∈ H1(Ω) to be the harmonic extension of uεk.

Our goal is to show that Uεk converges to an eigenfunctioncorresponding to Σk,β as ε→ 0.

20 / 23

StrategySuppose rε ∼ βε2.

For ε > 0, let uεk be a complete set of Stekov eigenfunctions

{∆uεk = 0 in Ω

ε

∂νuεk = σk

εuεk on ∂Ωε

∫∂Ωε|uk|2 = 1.

Define Uεk ∈ H1(Ω) to be the harmonic extension of uεk.

Our goal is to show that Uεk converges to an eigenfunctioncorresponding to Σk,β as ε→ 0.

20 / 23

StrategySuppose rε ∼ βε2.

For ε > 0, let uεk be a complete set of Stekov eigenfunctions

{∆uεk = 0 in Ω

ε

∂νuεk = σk

εuεk on ∂Ωε

∫∂Ωε|uk|2 = 1.

Define Uεk ∈ H1(Ω) to be the harmonic extension of uεk.

Our goal is to show that Uεk converges to an eigenfunctioncorresponding to Σk,β as ε→ 0.

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Let Uε = Uεk and σε = σεk.

The Dirichlet energy is bounded Known isoperimetric inequality∫Ω|∇Uε|2 ≤ Cσε ≤ C

Energy estimates on holes

WLOG Uε ⇀ U in H1(Ω) and σε → Σ.

∫Ω∇Uε · ∇V =

∫Ωε∪Tε

∇uε · ∇v = σε∫∂Ωε

uv +

→ 0︷ ︸︸ ︷∫Tε∇uε · ∇v

∫Ω∇U · ∇V − Σ

∫∂Ω

UV = Σ limε→0

∫∂Tε

uεV = 2πβΣ

∫ΩUV

Energy estimates on holes More work!

QED

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Let Uε = Uεk and σε = σεk.

The Dirichlet energy is bounded Known isoperimetric inequality∫Ω|∇Uε|2 ≤ Cσε ≤ C

Energy estimates on holes

WLOG Uε ⇀ U in H1(Ω) and σε → Σ.

∫Ω∇Uε · ∇V =

∫Ωε∪Tε

∇uε · ∇v = σε∫∂Ωε

uv +

→ 0︷ ︸︸ ︷∫Tε∇uε · ∇v

∫Ω∇U · ∇V − Σ

∫∂Ω

UV = Σ limε→0

∫∂Tε

uεV = 2πβΣ

∫ΩUV

Energy estimates on holes More work!

QED

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Let Uε = Uεk and σε = σεk.

The Dirichlet energy is bounded Known isoperimetric inequality∫Ω|∇Uε|2 ≤ Cσε ≤ C

Energy estimates on holes

WLOG Uε ⇀ U in H1(Ω) and σε → Σ.

∫Ω∇Uε · ∇V =

∫Ωε∪Tε

∇uε · ∇v = σε∫∂Ωε

uv +

→ 0︷ ︸︸ ︷∫Tε∇uε · ∇v

∫Ω∇U · ∇V − Σ

∫∂Ω

UV = Σ limε→0

∫∂Tε

uεV = 2πβΣ

∫ΩUV

Energy estimates on holes More work!

QED

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Let Uε = Uεk and σε = σεk.

The Dirichlet energy is bounded Known isoperimetric inequality∫Ω|∇Uε|2 ≤ Cσε ≤ C

Energy estimates on holes

WLOG Uε ⇀ U in H1(Ω) and σε → Σ.

∫Ω∇Uε · ∇V =

∫Ωε∪Tε

∇uε · ∇v = σε∫∂Ωε

uv +

→ 0︷ ︸︸ ︷∫Tε∇uε · ∇v

∫Ω∇U · ∇V − Σ

∫∂Ω

UV = Σ limε→0

∫∂Tε

uεV = 2πβΣ

∫ΩUV

Energy estimates on holes More work!

QED

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Thank you for your attention!

Homogenisation theory