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)
2 / 23
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)
2 / 23
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
4 / 23
2. Homogenization theory
5 / 23
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
6 / 23
Ωε = Ω \ Tε where Tε =⋃k∈Iε
B(εk, rε)
7 / 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
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
11 / 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
11 / 23
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)
12 / 23
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
12 / 23
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 ??
12 / 23
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
14 / 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
14 / 23
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|}
15 / 23
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|}
15 / 23
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|}
15 / 23
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
16 / 23
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
17 / 23
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
17 / 23
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. . .
18 / 23
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)
19 / 23
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)
19 / 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.
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
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
21 / 23
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
21 / 23
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
21 / 23
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
21 / 23
Thank you for your attention!
Homogenisation theory