Spectral analysis of a Neumannbiharmonic operator on a planar
dumbbell domain
Francesco Ferraresso(based on a joint work with J.M. Arrieta and P.D. Lamberti)
Workshop on geometric spectral theoryNeuchatel - 22.06.2017
The dumbbell
Ω
ΩRε
1
(a) Dumbbell domain.
Ω
ΩR0
1
(b) Limit domain.
2 of 19
The dumbbell
Ω
ΩRε
1
(a) Dumbbell domain.
Ω
ΩR0
1
(b) Limit domain.
2 of 19
Dumbbells in the literature
Spectral analysis on dumbbells is important in:
Geometric spectral theory→ Cheeger’s constant;
Parabolic reaction-diffusion equations→ Matano’scounterexample;
Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.
Many contributions in the case of the Laplace operator:
Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...
Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...
...and many others.
3 of 19
Dumbbells in the literature
Spectral analysis on dumbbells is important in:
Geometric spectral theory→ Cheeger’s constant;
Parabolic reaction-diffusion equations→ Matano’scounterexample;
Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.
Many contributions in the case of the Laplace operator:
Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...
Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...
...and many others.
3 of 19
Dumbbells in the literature
Spectral analysis on dumbbells is important in:
Geometric spectral theory→ Cheeger’s constant;
Parabolic reaction-diffusion equations→ Matano’scounterexample;
Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.
Many contributions in the case of the Laplace operator:
Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...
Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...
...and many others.
3 of 19
Dumbbells in the literature
Spectral analysis on dumbbells is important in:
Geometric spectral theory→ Cheeger’s constant;
Parabolic reaction-diffusion equations→ Matano’scounterexample;
Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.
Many contributions in the case of the Laplace operator:
Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...
Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...
...and many others.
3 of 19
Dumbbells in the literature
Spectral analysis on dumbbells is important in:
Geometric spectral theory→ Cheeger’s constant;
Parabolic reaction-diffusion equations→ Matano’scounterexample;
Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.
Many contributions in the case of the Laplace operator:
Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...
Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...
...and many others.
3 of 19
Dumbbells in the literature
Spectral analysis on dumbbells is important in:
Geometric spectral theory→ Cheeger’s constant;
Parabolic reaction-diffusion equations→ Matano’scounterexample;
Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.
Many contributions in the case of the Laplace operator:
Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...
Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...
...and many others.
3 of 19
Dumbbells in the literature
Spectral analysis on dumbbells is important in:
Geometric spectral theory→ Cheeger’s constant;
Parabolic reaction-diffusion equations→ Matano’scounterexample;
Spectral stability problems→ classical counterexample tospectral continuity of Neumann Laplacian eigenvalues.
Many contributions in the case of the Laplace operator:
Dirichlet boundary conditions: Abatangelo-Felli-Terracini,Davies, Taylor, ...
Neumann boundary conditions: Arrieta et al., Jimbo,Sanchez-Palencia, Morini-Slastikov,...
...and many others.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN.
Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω)
with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent,
itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete
λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε
in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
Motivations: eigenvalues and perturbations
Let Ω be a bounded smooth open set of RN. Given a nonnegativeself-adjoint elliptic operator H on L2(Ω) with compact resolvent, itsspectrum is discrete λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ . . . .
Problem: Provide (necessary and) sufficient conditions on Ωε , Ωand on the perturbation Ω 7→ Ωε in order to have the continuity ofthe maps ε 7→ λn[Ωε ] at ε = 0, for all n ∈ N.
There are many cases of quite regular perturbations providingcounterexamples
Aim: better understanding of this “pathological” cases.
3 of 19
The dumbbell problem
The dumbbell domains Ωε ⊂ R2 are perturbations of a fixed
bounded open set Ω with two (or more) connected components.
Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,
Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).
x
y
0 1
εg(0) Rεεg(1)
ΩL
ΩR
1
4 of 19
The dumbbell problemThe dumbbell domains Ωε ⊂ R
2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.
Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,
Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).
x
y
0 1
εg(0) Rεεg(1)
ΩL
ΩR
1
4 of 19
The dumbbell problemThe dumbbell domains Ωε ⊂ R
2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.
Ω := ΩL ∪ ΩR
Ωε := Ω ∪ Rε ,
Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).
x
y
0 1
εg(0) Rεεg(1)
ΩL
ΩR
1
4 of 19
The dumbbell problemThe dumbbell domains Ωε ⊂ R
2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.
Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,
Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).
x
y
0 1
εg(0) Rεεg(1)
ΩL
ΩR
1
4 of 19
The dumbbell problemThe dumbbell domains Ωε ⊂ R
2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.
Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,
Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).
x
y
0 1
εg(0) Rεεg(1)
ΩL
ΩR
1
4 of 19
The dumbbell problemThe dumbbell domains Ωε ⊂ R
2 are perturbations of a fixedbounded open set Ω with two (or more) connected components.
Ω := ΩL ∪ ΩR Ωε := Ω ∪ Rε ,
Rε = (x, y) ∈ R2 : 0 < x < 1, 0 < y < εg(x), g ∈ C2([0, 1]).
x
y
0 1
εg(0) Rεεg(1)
ΩL
ΩR
1
4 of 19
A dumbbell-shaped free plate
On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2
∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,
τ∂u∂n − (1 − σ) div∂Ωε (D
2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .
where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds
∫Ωε
(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)
∫Ωε
uψ dx
We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.
5 of 19
A dumbbell-shaped free plate
On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2
∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,
τ∂u∂n − (1 − σ) div∂Ωε (D
2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .
where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds
∫Ωε
(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)
∫Ωε
uψ dx
We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.
5 of 19
A dumbbell-shaped free plate
On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2
∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,
τ∂u∂n − (1 − σ) div∂Ωε (D
2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .
where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε).
The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds
∫Ωε
(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)
∫Ωε
uψ dx
We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.
5 of 19
A dumbbell-shaped free plate
On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2
∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,
τ∂u∂n − (1 − σ) div∂Ωε (D
2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .
where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis:
find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds
∫Ωε
(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)
∫Ωε
uψ dx
We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.
5 of 19
A dumbbell-shaped free plate
On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2
∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,
τ∂u∂n − (1 − σ) div∂Ωε (D
2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .
where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds
∫Ωε
(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)
∫Ωε
uψ dx
We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.
5 of 19
A dumbbell-shaped free plate
On the dumbbell Ωε ⊂ R2 we consider a Neumann problem for ∆2
∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(1 − σ)∂2u∂n2 + σ∆u = 0, on ∂Ωε ,
τ∂u∂n − (1 − σ) div∂Ωε (D
2u · n)∂Ωε −∂(∆u)∂n = 0, on ∂Ωε .
where σ ∈ (−1, 1), τ ≥ 0, and u ∈ H2(Ωε). The weak formulationis: find u ∈ H2(Ωε) such that, for all ψ ∈ H2(Ωε) there holds
∫Ωε
(1−σ)D2u : D2ψ+σ∆u∆ψ+τ∇u·∇ψ+uψ dx = λ(Ωε)
∫Ωε
uψ dx
We denote by (ϕεn, λn(Ωε)) the eigenpairs ∀n ∈ N.5 of 19
Spectral convergence
Definition
Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0
if
λn[Ωε ]→ λn[Ω0] for all n ∈ N
The spectral projections PΩεa converge to PΩ0
a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the
projections PΩεa from L2(RN) into L2(Ωε) by
PΩεa (ψ) =
∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that
sup
∥∥∥∥PΩεa (ψ) − PΩ0
a (ψ)∥∥∥∥
L2(Ω0)+
∥∥∥PΩεa (ψ)
∥∥∥L2(Ωε\Ω0)
→ 0,
where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.
6 of 19
Spectral convergence
Definition
Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if
λn[Ωε ]→ λn[Ω0] for all n ∈ N
The spectral projections PΩεa converge to PΩ0
a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the
projections PΩεa from L2(RN) into L2(Ωε) by
PΩεa (ψ) =
∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that
sup
∥∥∥∥PΩεa (ψ) − PΩ0
a (ψ)∥∥∥∥
L2(Ω0)+
∥∥∥PΩεa (ψ)
∥∥∥L2(Ωε\Ω0)
→ 0,
where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.
6 of 19
Spectral convergence
Definition
Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if
λn[Ωε ]→ λn[Ω0] for all n ∈ N
The spectral projections PΩεa converge to PΩ0
a in L2
i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the
projections PΩεa from L2(RN) into L2(Ωε) by
PΩεa (ψ) =
∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that
sup
∥∥∥∥PΩεa (ψ) − PΩ0
a (ψ)∥∥∥∥
L2(Ω0)+
∥∥∥PΩεa (ψ)
∥∥∥L2(Ωε\Ω0)
→ 0,
where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.
6 of 19
Spectral convergence
Definition
Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if
λn[Ωε ]→ λn[Ω0] for all n ∈ N
The spectral projections PΩεa converge to PΩ0
a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the
projections PΩεa from L2(RN) into L2(Ωε) by
PΩεa (ψ) =
∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ]
and we ask that
sup
∥∥∥∥PΩεa (ψ) − PΩ0
a (ψ)∥∥∥∥
L2(Ω0)+
∥∥∥PΩεa (ψ)
∥∥∥L2(Ωε\Ω0)
→ 0,
where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.
6 of 19
Spectral convergence
Definition
Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if
λn[Ωε ]→ λn[Ω0] for all n ∈ N
The spectral projections PΩεa converge to PΩ0
a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the
projections PΩεa from L2(RN) into L2(Ωε) by
PΩεa (ψ) =
∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that
sup
∥∥∥∥PΩεa (ψ) − PΩ0
a (ψ)∥∥∥∥
L2(Ω0)+
∥∥∥PΩεa (ψ)
∥∥∥L2(Ωε\Ω0)
→ 0,
where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.
6 of 19
Spectral convergence
Definition
Given the operator Tε on L2(Ωε), we say that Tε is spectrallyconverging to T0 on L2(Ω0), with Ω0 ⊂ Ωε for all ε > 0 if
λn[Ωε ]→ λn[Ω0] for all n ∈ N
The spectral projections PΩεa converge to PΩ0
a in L2 i.e., forfixed a ∈ R+ \ λj[Ω]∞j=0, λn[Ω] < a < λn+1[Ω] we define the
projections PΩεa from L2(RN) into L2(Ωε) by
PΩεa (ψ) =
∑ni=1(ui[Ωε ], ψ)L2(Ωε)ui[Ωε ] and we ask that
sup
∥∥∥∥PΩεa (ψ) − PΩ0
a (ψ)∥∥∥∥
L2(Ω0)+
∥∥∥PΩεa (ψ)
∥∥∥L2(Ωε\Ω0)
→ 0,
where the sup is on all ψ ∈ L2(RN) with ‖ψ‖L2(RN) = 1.
6 of 19
Spectral stability
We say that there is spectral stability
if the operator Tε := ∆2Ωε
associated with∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
spectrally converges to the operator T0 := ∆2Ω associated with∆2u − τ∆u + u = λn(Ω) u, in Ω,
(NBC)σ, on ∂Ω
7 of 19
Spectral stability
We say that there is spectral stability if the operator Tε := ∆2Ωε
associated with∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
spectrally converges to the operator T0 := ∆2Ω associated with∆2u − τ∆u + u = λn(Ω) u, in Ω,
(NBC)σ, on ∂Ω
7 of 19
Spectral stability
We say that there is spectral stability if the operator Tε := ∆2Ωε
associated with∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
spectrally converges to the operator T0 := ∆2Ω
associated with∆2u − τ∆u + u = λn(Ω) u, in Ω,
(NBC)σ, on ∂Ω
7 of 19
Spectral stability
We say that there is spectral stability if the operator Tε := ∆2Ωε
associated with∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
spectrally converges to the operator T0 := ∆2Ω associated with∆2u − τ∆u + u = λn(Ω) u, in Ω,
(NBC)σ, on ∂Ω
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Characterization of the spectral stability
General condition for bounded open sets Ωε , Ω.
Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that
|Ω \ Kε | → 0, as ε → 0
If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then
limε→0‖vε‖L2(Ωε\Kε) = 0
Arrieta’s condition⇒ Spectral convergence
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Characterization of the spectral stability
General condition for bounded open sets Ωε , Ω.
Arrieta’s condition:
There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that
|Ω \ Kε | → 0, as ε → 0
If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then
limε→0‖vε‖L2(Ωε\Kε) = 0
Arrieta’s condition⇒ Spectral convergence
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Characterization of the spectral stability
General condition for bounded open sets Ωε , Ω.
Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that
|Ω \ Kε | → 0, as ε → 0
If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then
limε→0‖vε‖L2(Ωε\Kε) = 0
Arrieta’s condition⇒ Spectral convergence
8 of 19
Characterization of the spectral stability
General condition for bounded open sets Ωε , Ω.
Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that
|Ω \ Kε | → 0, as ε → 0
If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then
limε→0‖vε‖L2(Ωε\Kε) = 0
Arrieta’s condition⇒ Spectral convergence
8 of 19
Characterization of the spectral stability
General condition for bounded open sets Ωε , Ω.
Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that
|Ω \ Kε | → 0, as ε → 0
If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then
limε→0‖vε‖L2(Ωε\Kε) = 0
Arrieta’s condition⇒ Spectral convergence
8 of 19
Characterization of the spectral stability
General condition for bounded open sets Ωε , Ω.
Arrieta’s condition:There exist uniformly Lipschitz open sets Kε ⊂ Ω ∩ Ωε such that
|Ω \ Kε | → 0, as ε → 0
If vε ∈ W2,2(Ωε) and supε>0‖vε‖W2,2(Ωε) < ∞ then
limε→0‖vε‖L2(Ωε\Kε) = 0
Arrieta’s condition⇒ Spectral convergence
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Spectral stability II
In our case, Arrieta’s condition is related to the limit as ε → 0 of
τε = infφε ∈W2,2(Ωε)φε=0 in Ω
∫Ωε
(1 − σ)∣∣∣D2φε
∣∣∣2 + σ |∆φε |2 + τ |∇φε |
2 + |φε |2 dx
‖φε‖2L2(Rε)
Arrieta’s condition is equivalent to limε→0 τε = +∞.
The dumbbell Ωε violates this condition!
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Spectral stability II
In our case, Arrieta’s condition is related to the limit as ε → 0 of
τε = infφε ∈W2,2(Ωε)φε=0 in Ω
∫Ωε
(1 − σ)∣∣∣D2φε
∣∣∣2 + σ |∆φε |2 + τ |∇φε |
2 + |φε |2 dx
‖φε‖2L2(Rε)
Arrieta’s condition is equivalent to limε→0 τε = +∞.
The dumbbell Ωε violates this condition!
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Spectral stability II
In our case, Arrieta’s condition is related to the limit as ε → 0 of
τε = infφε ∈W2,2(Ωε)φε=0 in Ω
∫Ωε
(1 − σ)∣∣∣D2φε
∣∣∣2 + σ |∆φε |2 + τ |∇φε |
2 + |φε |2 dx
‖φε‖2L2(Rε)
Arrieta’s condition is equivalent to limε→0 τε = +∞.
The dumbbell Ωε violates this condition!
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Spectral instability
Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a
Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:
‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1
or equivalently, τε < ∞.
Concentration phenomenon⇒ Spectral instability
There must be extra eigenvalues in the limit!
10 of 19
Spectral instability
Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a
Concentration phenomenon:
it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:
‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1
or equivalently, τε < ∞.
Concentration phenomenon⇒ Spectral instability
There must be extra eigenvalues in the limit!
10 of 19
Spectral instability
Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a
Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:
‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1
or equivalently, τε < ∞.
Concentration phenomenon⇒ Spectral instability
There must be extra eigenvalues in the limit!
10 of 19
Spectral instability
Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a
Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:
‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1
or equivalently, τε < ∞.
Concentration phenomenon⇒ Spectral instability
There must be extra eigenvalues in the limit!
10 of 19
Spectral instability
Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a
Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:
‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1
or equivalently, τε < ∞.
Concentration phenomenon⇒ Spectral instability
There must be extra eigenvalues in the limit!
10 of 19
Spectral instability
Recall that Rε is collapsing to a lower dimensional manifold asε → 0 (Rε is a thin domain). In the dumbbell there will be a
Concentration phenomenon: it appears when the eigenfunctionsuε have L2-norm that concentrates on Rε . More precisely:
‖uε‖H2(Ωε) ≤ C , ‖uε‖L2(Rε) → 1
or equivalently, τε < ∞.
Concentration phenomenon⇒ Spectral instability
There must be extra eigenvalues in the limit!
10 of 19
Main problem
Question: can we characterize these extra eigenvalues?
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Two auxiliary problems
We introduce the eigenpairs (ϕΩk , ωk ) of∆2w − τ∆w + w = ωk w, in Ω
(NBC)σ, on ∂Ω,
and the eigenpairs (γεl , θεl ) of
∆2v − τ∆v + v = θεl v , in Rε
(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε
τ∂v∂n − (1 − σ) divΓε (D
2v · n)Γε −∂(∆v)∂n = 0, on Γε ,
v = 0 = ∂v∂n , on Lε .
where Γε = (x, y) : 0 < x < 1, y = εg(x) ∪ (x, 0) : 0 < x < 1and Lε = ∂Rε \ Γε .
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Two auxiliary problems
We introduce the eigenpairs (ϕΩk , ωk ) of∆2w − τ∆w + w = ωk w, in Ω
(NBC)σ, on ∂Ω,
and the eigenpairs (γεl , θεl ) of
∆2v − τ∆v + v = θεl v , in Rε
(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε
τ∂v∂n − (1 − σ) divΓε (D
2v · n)Γε −∂(∆v)∂n = 0, on Γε ,
v = 0 = ∂v∂n , on Lε .
where Γε = (x, y) : 0 < x < 1, y = εg(x) ∪ (x, 0) : 0 < x < 1and Lε = ∂Rε \ Γε .
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x
y
0
Γε
Lε
1
εg(0) εg(1)
1
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An additional assumption
Note that the dumbbell Ωε is not smooth on the junctions, i.e., on∂Rε ∩ ∂Ω.
We need some condition to control the eigenfunctionsnear the junctions.This condition is usually called H-Condition:
x
y
0 1
εg(0) εg(1)
1
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An additional assumption
Note that the dumbbell Ωε is not smooth on the junctions, i.e., on∂Rε ∩ ∂Ω. We need some condition to control the eigenfunctionsnear the junctions.
This condition is usually called H-Condition:
x
y
0 1
εg(0) εg(1)
1
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An additional assumption
Note that the dumbbell Ωε is not smooth on the junctions, i.e., on∂Rε ∩ ∂Ω. We need some condition to control the eigenfunctionsnear the junctions.This condition is usually called H-Condition:
x
y
0 1
εg(0) εg(1)
1
12 of 19
An additional assumption
Note that the dumbbell Ωε is not smooth on the junctions, i.e., on∂Rε ∩ ∂Ω. We need some condition to control the eigenfunctionsnear the junctions.This condition is usually called H-Condition:
x
y
0 1
εg(0) εg(1)
1
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Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
and that ωk and θεl are the eigenvalues of
∆2w − τ∆w + w = ωk w, in Ω
(NBC)σ, on ∂Ω
∆2v − τ∆v + v = θεl v , in Rε
(NBC)σ, on Γε
v = 0 = ∂v∂n , on Lε
Define (λεn)n = (ωk )k ∪ (θεl )l . Assume that Rε satisfies theH-condition. Then
|λn(Ωε) − λεn | → 0, as ε → 0.
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Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
and that ωk and θεl are the eigenvalues of
∆2w − τ∆w + w = ωk w, in Ω
(NBC)σ, on ∂Ω
∆2v − τ∆v + v = θεl v , in Rε
(NBC)σ, on Γε
v = 0 = ∂v∂n , on Lε
Define (λεn)n = (ωk )k ∪ (θεl )l . Assume that Rε satisfies theH-condition. Then
|λn(Ωε) − λεn | → 0, as ε → 0.
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Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
and that ωk and θεl are the eigenvalues of
∆2w − τ∆w + w = ωk w, in Ω
(NBC)σ, on ∂Ω
∆2v − τ∆v + v = θεl v , in Rε
(NBC)σ, on Γε
v = 0 = ∂v∂n , on Lε
Define (λεn)n = (ωk )k ∪ (θεl )l .
Assume that Rε satisfies theH-condition. Then
|λn(Ωε) − λεn | → 0, as ε → 0.
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Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
and that ωk and θεl are the eigenvalues of
∆2w − τ∆w + w = ωk w, in Ω
(NBC)σ, on ∂Ω
∆2v − τ∆v + v = θεl v , in Rε
(NBC)σ, on Γε
v = 0 = ∂v∂n , on Lε
Define (λεn)n = (ωk )k ∪ (θεl )l . Assume that Rε satisfies theH-condition.
Then
|λn(Ωε) − λεn | → 0, as ε → 0.
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Asymptotic spectral decompositionRecall that λn(Ωε) are the eigenvalues of∆2u − τ∆u + u = λn(Ωε) u, in Ωε ,
(NBC)σ, on ∂Ωε
and that ωk and θεl are the eigenvalues of
∆2w − τ∆w + w = ωk w, in Ω
(NBC)σ, on ∂Ω
∆2v − τ∆v + v = θεl v , in Rε
(NBC)σ, on Γε
v = 0 = ∂v∂n , on Lε
Define (λεn)n = (ωk )k ∪ (θεl )l . Assume that Rε satisfies theH-condition. Then
|λn(Ωε) − λεn | → 0, as ε → 0.
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A difficult problem
At this point we are interested in the convergence of θεl as ε → 0.
We would like to pass directly to the limit in the PDE problem∆2v − τ∆v + v = θεl v , in Rε
(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε
τ∂v∂n − (1 − σ) divΓε (D
2v · n)Γε −∂(∆v)∂n = 0, on Γε ,
v = 0 = ∂v∂n , on Lε .
but this seems a tough problem. We took a different path.
14 of 19
A difficult problem
At this point we are interested in the convergence of θεl as ε → 0.We would like to pass directly to the limit in the PDE problem
∆2v − τ∆v + v = θεl v , in Rε
(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε
τ∂v∂n − (1 − σ) divΓε (D
2v · n)Γε −∂(∆v)∂n = 0, on Γε ,
v = 0 = ∂v∂n , on Lε .
but this seems a tough problem. We took a different path.
14 of 19
A difficult problem
At this point we are interested in the convergence of θεl as ε → 0.We would like to pass directly to the limit in the PDE problem
∆2v − τ∆v + v = θεl v , in Rε
(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε
τ∂v∂n − (1 − σ) divΓε (D
2v · n)Γε −∂(∆v)∂n = 0, on Γε ,
v = 0 = ∂v∂n , on Lε .
but this seems a tough problem.
We took a different path.
14 of 19
A difficult problem
At this point we are interested in the convergence of θεl as ε → 0.We would like to pass directly to the limit in the PDE problem
∆2v − τ∆v + v = θεl v , in Rε
(1 − σ)∂2v∂n2 + σ∆v = 0, on Γε
τ∂v∂n − (1 − σ) divΓε (D
2v · n)Γε −∂(∆v)∂n = 0, on Γε ,
v = 0 = ∂v∂n , on Lε .
but this seems a tough problem. We took a different path.
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Compact convergence approach
Proof of the convergence to a suitable one-dimensional problemvia:
Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;
Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];
Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;
Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.
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Compact convergence approach
Proof of the convergence to a suitable one-dimensional problemvia:
Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;
Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];
Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;
Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.
15 of 19
Compact convergence approach
Proof of the convergence to a suitable one-dimensional problemvia:
Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;
Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];
Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;
Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.
15 of 19
Compact convergence approach
Proof of the convergence to a suitable one-dimensional problemvia:
Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;
Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];
Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;
Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.
15 of 19
Compact convergence approach
Proof of the convergence to a suitable one-dimensional problemvia:
Rescaling of the problem in Rε to a problem for a perturbeddifferential operator in the fixed domain R1;
Homogenization/thin domain techniques in order to find thelimiting differential problem in the segment [0,1];
Compact convergence of the resolvent operators associatedwith the rescaled problem to the resolvent operator of thelimiting problem;
Abstract result: compact convergence of resolvent operatorsimplies spectral convergence.
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The limit problem
One can prove that∆2v − τ∆v + v = θεl v , in Rε
(NBC)σ, on Γε ,
v = 0 = ∂v∂n , on Lε .
compact converges to a suitable rescaling of1−σ2
g (gv′′)′′ − τg (gv′)′ + v = θlv , in (0, 1)
v(0) = v(1) = 0, v′(0) = v′(1) = 0
We denote by(hi , θi) the eigenpairs associated with this ODE.
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The limit problem
One can prove that∆2v − τ∆v + v = θεl v , in Rε
(NBC)σ, on Γε ,
v = 0 = ∂v∂n , on Lε .
compact converges to a suitable rescaling of1−σ2
g (gv′′)′′ − τg (gv′)′ + v = θlv , in (0, 1)
v(0) = v(1) = 0, v′(0) = v′(1) = 0
We denote by(hi , θi) the eigenpairs associated with this ODE.
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The limit problem
One can prove that∆2v − τ∆v + v = θεl v , in Rε
(NBC)σ, on Γε ,
v = 0 = ∂v∂n , on Lε .
compact converges to a suitable rescaling of1−σ2
g (gv′′)′′ − τg (gv′)′ + v = θlv , in (0, 1)
v(0) = v(1) = 0, v′(0) = v′(1) = 0
We denote by(hi , θi) the eigenpairs associated with this ODE.
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Notation
Define the operator Eε : H2(0, 1)→ H2(Rε) by
Eεv(x, y) = v(x)
for all (x, y) ∈ Rε . Moreover let N(·) be the counting functiondefined by
N(x) = #λi : i ∈ N, λi ≤ x
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Final convergence result
Theorem (Arrieta, F., Lamberti, 2016)
Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.
The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then
∥∥∥ϕεn |Ω∥∥∥L2(Ω)
→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑
i=1
(ϕεn, ϕΩi )L2(Ω)ϕ
Ωi
∥∥∥∥∥∥∥∥H2(Ω)
→ 0
as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −
N(θl)∑i=1
(ϕεn, ε−1/2Eεhi)L2(Rε)ε
−1/2Eεhi
∥∥∥∥∥∥∥∥L2(Rε)
→ 0
as ε → 0.
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Final convergence result
Theorem (Arrieta, F., Lamberti, 2016)
Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.
The eigenvalues λn(Ωε) converge either to ωk
or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then
∥∥∥ϕεn |Ω∥∥∥L2(Ω)
→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑
i=1
(ϕεn, ϕΩi )L2(Ω)ϕ
Ωi
∥∥∥∥∥∥∥∥H2(Ω)
→ 0
as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −
N(θl)∑i=1
(ϕεn, ε−1/2Eεhi)L2(Rε)ε
−1/2Eεhi
∥∥∥∥∥∥∥∥L2(Rε)
→ 0
as ε → 0.
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Final convergence result
Theorem (Arrieta, F., Lamberti, 2016)
Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.
The eigenvalues λn(Ωε) converge either to ωk or to θl .
Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then
∥∥∥ϕεn |Ω∥∥∥L2(Ω)
→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑
i=1
(ϕεn, ϕΩi )L2(Ω)ϕ
Ωi
∥∥∥∥∥∥∥∥H2(Ω)
→ 0
as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −
N(θl)∑i=1
(ϕεn, ε−1/2Eεhi)L2(Rε)ε
−1/2Eεhi
∥∥∥∥∥∥∥∥L2(Rε)
→ 0
as ε → 0.
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Final convergence result
Theorem (Arrieta, F., Lamberti, 2016)
Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.
The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then
∥∥∥ϕεn |Ω∥∥∥L2(Ω)
→ 1,
and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑
i=1
(ϕεn, ϕΩi )L2(Ω)ϕ
Ωi
∥∥∥∥∥∥∥∥H2(Ω)
→ 0
as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −
N(θl)∑i=1
(ϕεn, ε−1/2Eεhi)L2(Rε)ε
−1/2Eεhi
∥∥∥∥∥∥∥∥L2(Rε)
→ 0
as ε → 0.
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Final convergence result
Theorem (Arrieta, F., Lamberti, 2016)
Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.
The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then
∥∥∥ϕεn |Ω∥∥∥L2(Ω)
→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑
i=1
(ϕεn, ϕΩi )L2(Ω)ϕ
Ωi
∥∥∥∥∥∥∥∥H2(Ω)
→ 0
as ε → 0.
Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −
N(θl)∑i=1
(ϕεn, ε−1/2Eεhi)L2(Rε)ε
−1/2Eεhi
∥∥∥∥∥∥∥∥L2(Rε)
→ 0
as ε → 0.
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Final convergence result
Theorem (Arrieta, F., Lamberti, 2016)
Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.
The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then
∥∥∥ϕεn |Ω∥∥∥L2(Ω)
→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑
i=1
(ϕεn, ϕΩi )L2(Ω)ϕ
Ωi
∥∥∥∥∥∥∥∥H2(Ω)
→ 0
as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω)
and∥∥∥∥∥∥∥∥ϕεn −N(θl)∑i=1
(ϕεn, ε−1/2Eεhi)L2(Rε)ε
−1/2Eεhi
∥∥∥∥∥∥∥∥L2(Rε)
→ 0
as ε → 0.
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Final convergence result
Theorem (Arrieta, F., Lamberti, 2016)
Let Ωε ⊂ R2 be a dumbbell domain satisfying the (H)-condition.
The eigenvalues λn(Ωε) converge either to ωk or to θl . Moreover, ifλn(Ωε)→ ωk for some k ∈ N, then
∥∥∥ϕεn |Ω∥∥∥L2(Ω)
→ 1, and∥∥∥∥∥∥∥∥ϕεn |Ω −N(ωk )∑
i=1
(ϕεn, ϕΩi )L2(Ω)ϕ
Ωi
∥∥∥∥∥∥∥∥H2(Ω)
→ 0
as ε → 0. Otherwise, if λn(Ωε)→ θl for some l ∈ N, then ϕεn |Ω → 0in L2(Ω) and∥∥∥∥∥∥∥∥ϕεn −
N(θl)∑i=1
(ϕεn, ε−1/2Eεhi)L2(Rε)ε
−1/2Eεhi
∥∥∥∥∥∥∥∥L2(Rε)
→ 0
as ε → 0.18 of 19
Principal references
S. Jimbo, “The singularly perturbed domain and the characterizationfor the eigenfunctions with Neumann boundary conditions”,J.Differential Equations, 77, 1989, 322-350.
J. M. Arrieta, “Neumann eigenvalue problems on exteriorperturbations of the domain”, J. Differential Equations, 117, 1995.
J. M. Arrieta, A. N. Carvalho, G. Losada-Cruz, “Dynamics indumbbell domains I. Continuity of the set of equilibria”,J. Differential Equations, 231 (2), 2006, 551-597.
J. M. Arrieta, F.F., P.D. Lamberti, “Spectral analysis of thebiharmonic operator subject to Neumann boundary conditions ondumbbell domains”, submitted, 2017
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Thank youfor your attention
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Open questions
1. Identification of the limit behavior of the eigenfunction ϕεn in[0, 1] when λn(Ωε)→ ωk .
2. Replacement of the L2(Rε) norm with a stronger Sobolevnorm in the final convergence result.
Partial results are available in the case σ = 0, the case σ , 0being open at the moment.
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Open questions
1. Identification of the limit behavior of the eigenfunction ϕεn in[0, 1] when λn(Ωε)→ ωk .
2. Replacement of the L2(Rε) norm with a stronger Sobolevnorm in the final convergence result.
Partial results are available in the case σ = 0, the case σ , 0being open at the moment.
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Open questions
1. Identification of the limit behavior of the eigenfunction ϕεn in[0, 1] when λn(Ωε)→ ωk .
2. Replacement of the L2(Rε) norm with a stronger Sobolevnorm in the final convergence result.
Partial results are available in the case σ = 0, the case σ , 0being open at the moment.
19 of 19