Asymptotics of Robin eigenvalues in domains with corners Magda - - PowerPoint PPT Presentation

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Asymptotics of Robin eigenvalues in domains with corners

Magda Khalile

Institut f¨ ur Analysis, Leibniz Universit¨ at Hannover

Joint work with T. Ourmi` eres-Bonafos (Dauphine) and K. Pankrashkin (Orsay)

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The Robin eigenvalue problem

Let Ω ⊂ Rd, d ≥ 2, be Lipschitz and bounded. For γ > 0, consider −∆u = Eu on Ω, ∂u ∂ν = γu on ∂Ω.

Ω ν

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The Robin eigenvalue problem

Let Ω ⊂ Rd, d ≥ 2, be Lipschitz and bounded. For γ > 0, consider −∆u = Eu on Ω, ∂u ∂ν = γu on ∂Ω.

Ω ν

• Behavior of the eigenvalues E as γ → +∞ ?

2/24

The Robin eigenvalue problem

Let Ω ⊂ Rd, d ≥ 2, be Lipschitz and bounded. For γ > 0, consider −∆u = Eu on Ω, ∂u ∂ν = γu on ∂Ω.

Ω ν

• Behavior of the eigenvalues E as γ → +∞ ?
• Influence of the geometry of Ω on the asymptotics ?

2/24

The Robin eigenvalue problem

Let Ω ⊂ Rd, d ≥ 2, be Lipschitz and bounded. For γ > 0, consider −∆u = Eu on Ω, ∂u ∂ν = γu on ∂Ω.

Ω ν

• Behavior of the eigenvalues E as γ → +∞ ?
• Influence of the geometry of Ω on the asymptotics ?

Variational approach: the Robin Laplacian Qγ

Ω is the unique self-adjoint operator

in L2(Ω) associated with the closed form qγ

Ω(u, u) =

• Ω

|∇u|2 − γ

• ∂Ω

|u|2ds, u ∈ H1(Ω).

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The Robin eigenvalue problem

Let Ω ⊂ Rd, d ≥ 2, be Lipschitz and bounded. For γ > 0, consider −∆u = Eu on Ω, ∂u ∂ν = γu on ∂Ω.

Ω ν

• Behavior of the eigenvalues E as γ → +∞ ?
• Influence of the geometry of Ω on the asymptotics ?

Variational approach: the Robin Laplacian Qγ

Ω is the unique self-adjoint operator

in L2(Ω) associated with the closed form qγ

Ω(u, u) =

• Ω

|∇u|2 − γ

• ∂Ω

|u|2ds, u ∈ H1(Ω).

◮ Let n ∈ N be fixed: En(Qγ

Ω) −

− − − →

γ→+∞ ?

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Motivation: link with Dirichlet Laplacians acting on domains collapsing

• n metric graphs (1/2)
• The min-max principle gives: En(Qγ

Ω) → −∞ as γ → +∞, in particular

En(Qγ

Ω) < 0 for γ large enough.

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Motivation: link with Dirichlet Laplacians acting on domains collapsing

• n metric graphs (1/2)
• The min-max principle gives: En(Qγ

Ω) → −∞ as γ → +∞, in particular

En(Qγ

Ω) < 0 for γ large enough.

• Estimate on the negative eigenvalues → Dirichlet-Neumann bracketing:

Ωǫ := {x ∈ Ω : dist(x, ∂Ω) < ǫ} with ǫ := ǫ(γ) such that ǫ → 0 as γ → +∞.

ǫ

≤ ≤

Robin Neumann Robin Ωǫ Ω ǫ Robin Dirichlet Ωǫ

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Motivation: link with Dirichlet Laplacians acting on domains collapsing

• n metric graphs (1/2)
• The min-max principle gives: En(Qγ

Ω) → −∞ as γ → +∞, in particular

En(Qγ

Ω) < 0 for γ large enough.

• Estimate on the negative eigenvalues → Dirichlet-Neumann bracketing:

Ωǫ := {x ∈ Ω : dist(x, ∂Ω) < ǫ} with ǫ := ǫ(γ) such that ǫ → 0 as γ → +∞.

ǫ

≤ ≤

Robin Neumann Robin Ωǫ Ω ǫ Robin Dirichlet Ωǫ

• Geometrically the problem share some links with Dirichlet Laplacians acting on

domains collapsing on metric graphs [Grieser, 2008; Molchanov-Vainberg, 2007; Post,

2005]

Ωε Γ ε ε → 0

−∆u = En(ǫ)u on Ωǫ, u = 0 on ∂Ωǫ. ◮ Let n ∈ N be fixed: En(ǫ) − − − − →

ǫ→+0 ?

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Motivation: link with Dirichlet Laplacians acting on domains collapsing

• n metric graphs (2/2)

By rescaling, the limit object Ω0 := limǫ→0 Ωǫ is:

Ω0

−∆D

0 := Dirichlet Laplacian on Ω0,

N0 := # specdisc(−∆D

0 ) < +∞,

ν := inf specess(−∆D

0 ).

Asymptotics of Dirichlet eigenvalues on collapsing domains

[Grieser, 2008; Molchanov-Vainberg, 2007; Post, 2005]

There exists N ≥ N0 such that for ǫ → 0 there holds (i) for n ∈ {1, ..., N}, En(ǫ) = τn ǫ2 + O(e−c/ǫ), τn ∈ (0, ν], c > 0. (ii) for any j ∈ N, EN+j(ǫ) = ν ǫ2 + µj + O(ǫ), where the µj are the eigenvalues of a quantum graph Laplacian acting on Γ with transmission conditions at the vertices.

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Asymptotics of the Robin eigenvalues on smooth domains

[Exner-Minakov-Parnovski, 2014; Exner-Minakov, 2014; Pankrashkin-Popoff, 2015;...]

Effective operator [Pankrashkin-Popoff, 2016]

If ∂Ω is C2, then for any fixed n ∈ N, En(Qγ

Ω) = −γ2 + En(−∆∂Ω − γK) + O(log γ), as γ → +∞,

where −∆∂Ω is the Laplace-Beltrami operator acting in L2(∂Ω, ds) and K denotes the sum of the principal curvatures of ∂Ω.

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Asymptotics of the Robin eigenvalues on smooth domains

[Exner-Minakov-Parnovski, 2014; Exner-Minakov, 2014; Pankrashkin-Popoff, 2015;...]

Effective operator [Pankrashkin-Popoff, 2016]

If ∂Ω is C2, then for any fixed n ∈ N, En(Qγ

Ω) = −γ2 + En(−∆∂Ω − γK) + O(log γ), as γ → +∞,

where −∆∂Ω is the Laplace-Beltrami operator acting in L2(∂Ω, ds) and K denotes the sum of the principal curvatures of ∂Ω. → The study of −∆∂Ω − γK gives more precise asymptotics.

• If ∂Ω if C3 then En(Qγ

Ω) = −γ2 − γKmax + o(γ) where Kmax := max∂Ω K(s).

• Planar domains admitting a unique non-degenerate point of maximal curvature

[Helffer-Kachmar, 2017]:

• the eigenfunctions are localized near the maximum,
• complete asymptotic expansion of the eigenvalues.
• Smooth domains with exactly two points of maximal curvature

[Helffer-Kachmar-Raymond, 2017] : tunneling effect induced by the geometry.

• Weyl asymptotics [Kachmar-Keraval-Raymond, 2016] : asymptotic distribution of

the negative eigenvalues of Qγ

Ω.

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The corner domains

An example : the cube Cd ⊂ Rd

By separation of variables we have E1(Qγ

Cd) = −dγ2 + o(γ2), as γ → +∞.

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The corner domains

An example : the cube Cd ⊂ Rd

By separation of variables we have E1(Qγ

Cd) = −dγ2 + o(γ2), as γ → +∞.

Definition : the corner domains [Dauge, 1988; Grisvard, 1985]

Ω ⊂ Rd Lipschitz, piecewise smooth, bounded, and for any y ∈ ∂Ω, there exists V(y) : V(y) ∼ Uy ∩ B(0, r), r > 0, where Uy ⊂ Rd is a cone. In R2: → corner domains := curvilinear polygons, → tangent cones := infinite sectors.

y x

Ω

Uy Ux

2α 2α

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Asymptotics of the first Robin eigenvalue on corner domains

If Ω ⊂ Rd is a corner domain we denote by Uy the tangent cone of ∂Ω in y and Λ(Uy, γ) := inf spec(Qγ

Uy),

Λ(Uy, γ) = γ2Λ(Uy, 1).

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Asymptotics of the first Robin eigenvalue on corner domains

If Ω ⊂ Rd is a corner domain we denote by Uy the tangent cone of ∂Ω in y and Λ(Uy, γ) := inf spec(Qγ

Uy),

Λ(Uy, γ) = γ2Λ(Uy, 1).

First order asymptotics [Levitin-Parnovski, 2008; Bruneau-Popoff, 2016]

If Ω ⊂ Rd is a corner domains, then E1(Qγ

Ω) = CΩγ2 + o(γ2) as γ → +∞ with

CΩ := inf

y∈∂Ω Λ(Uy, 1).

→ The Robin Laplacians on the tangent cones determine the asymptotics of E1(Qγ

Ω):

they play the role of model operators.

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Asymptotics of the first Robin eigenvalue on corner domains

If Ω ⊂ Rd is a corner domain we denote by Uy the tangent cone of ∂Ω in y and Λ(Uy, γ) := inf spec(Qγ

Uy),

Λ(Uy, γ) = γ2Λ(Uy, 1).

First order asymptotics [Levitin-Parnovski, 2008; Bruneau-Popoff, 2016]

If Ω ⊂ Rd is a corner domains, then E1(Qγ

Ω) = CΩγ2 + o(γ2) as γ → +∞ with

CΩ := inf

y∈∂Ω Λ(Uy, 1).

→ The Robin Laplacians on the tangent cones determine the asymptotics of E1(Qγ

Ω):

they play the role of model operators. → In R2, the model operators are the Robin Laplacians acting on infinite sectors.

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• I. Study of the model operators : The Robin Laplacians on infinite sectors
• Essential spectrum ? Existence of discrete spectrum ?
• Properties of the associated eigenfunctions ?

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Study of the model operators (1/2)

O x2 x1 α α

Uα

ν

Infinite sector of half-aperture α ∈ (0, π) : Uα := {(x1, x2) ∈ R2 : |arg(x1 + ix2)| < α}. Robin Laplacian acting on L2(Uα) : Tα := Q1

Uα.

First properties:

[Levitin-Parnovski, 2008; Bruneau-Popoff, 2016; Kh.-Pankrashkin, 2018] For any α ∈ (0, π), specess(Tα) = [−1, +∞).

• If α ≥ π

2 , spec(Tα) = [−1, +∞).

• If α < π

2 , E1(Tα) = −

1 sin2 α associated to ϕα(x) = exp(− x1

sin α).

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Study of the model operators (2/2)

Eigenvalue counting function: Nα := #{n ∈ N : En(Tα) < −1}.

Finiteness of the discrete spectrum and behavior as α → 0 [Kh.-Pankrashkin, 2018]

• Nα < +∞ for any α > 0.
• (0, π

2 ) ∋ α → Nα is non-increasing.

• N π

6 = 1 and hence for any α ∈ [ π

6 , π 2 ), Nα = 1.

• Nα → +∞ as α → 0.

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Study of the model operators (2/2)

Eigenvalue counting function: Nα := #{n ∈ N : En(Tα) < −1}.

Finiteness of the discrete spectrum and behavior as α → 0 [Kh.-Pankrashkin, 2018]

• Nα < +∞ for any α > 0.
• (0, π

2 ) ∋ α → Nα is non-increasing.

• N π

6 = 1 and hence for any α ∈ [ π

6 , π 2 ), Nα = 1.

• Nα → +∞ as α → 0.

Localization of the eigenfunctions [Kh.-Pankrashkin, 2018]

If E < −1 is a discrete eigenvalue of Tα and u is an associated eigenfunction then for any ε ∈ (0, 1),

• Uα
• |∇u|2 + |u|2

e2(1−ε)√−1−E|x|dx < +∞. → the eigenfunctions are localized near the vertex of Uα → they decrease exponentially outside a neighborhood of the vertex

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• II. The corner induced Robin eigenvalues of curvilinear polygons
• First order asymptotics of the first eigenvalues created by the corners
• Localization of the associated eigenfunctions

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Eigenvalues created by the corners of the curvilinear polygons: notation

P := curvilinear polygon V := {convex corners of P} αv := half-aperture at v ∈ V, αv ∈ (0, π 2 )

y x

Ω

Uy Ux

2α 2α

Model operator:

T ⊕ :=

• v∈V

Tαv in

• v∈V

L2(Uαv) specess(T ⊕) = [−1, +∞), N ⊕ :=

• v∈V

Nαv < +∞

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Eigenvalues created by the corners of the curvilinear polygons: first

• rder asymptotics

Theorem [Kh., 2018]

For any n ∈ {1, ..., N ⊕}, En(Qγ

P) = En(T ⊕)γ2 + R(γ), as γ → +∞,

with R(γ) = O(e−cγ), c > 0, for a polygon with straight edges, R(γ) = O(γ

4 3 ) for a curvilinear polygon.

→ The associated eigenfunctions are localized near the convex vertices (Agmon estimate).

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Eigenvalues created by the corners of the curvilinear polygons: first

• rder asymptotics

Theorem [Kh., 2018]

For any n ∈ {1, ..., N ⊕}, En(Qγ

P) = En(T ⊕)γ2 + R(γ), as γ → +∞,

with R(γ) = O(e−cγ), c > 0, for a polygon with straight edges, R(γ) = O(γ

4 3 ) for a curvilinear polygon.

→ The associated eigenfunctions are localized near the convex vertices (Agmon estimate). Idea of the proof: construction of quasimodes by localizing the eigenfunctions of the model operator + partition of unity.

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Comments on the first order asymptotics

• If P has L ≥ 1 convex vertices with half-aperture π

6 ≤ α1 ≤ ... ≤ αL < π 2 , then En(Qγ

P) = −

γ2 sin2 αn + O(γ

4 3 ), as γ → +∞,

for any n = 1, ..., L.

• Proof. Each corner creates a single eigenvalue:

Nαn = 1 and E1(Tαn) = − 1 sin2 αn , n = 1, ..., L.

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Comments on the first order asymptotics

• If P has L ≥ 1 convex vertices with half-aperture π

6 ≤ α1 ≤ ... ≤ αL < π 2 , then En(Qγ

P) = −

γ2 sin2 αn + O(γ

4 3 ), as γ → +∞,

for any n = 1, ..., L.

• Proof. Each corner creates a single eigenvalue:

Nαn = 1 and E1(Tαn) = − 1 sin2 αn , n = 1, ..., L.

• Tunneling effect created by equal angles:

◮ Isosceles triangles [Helffer-Pankrashkin, 2015] : asymptotics of E2(Qγ

P) − E1(Qγ P) as γ → +∞.

◮ Squares (separation of variables): if ℓ is a square of length ℓ then E1(Qγ

ℓ) = −2γ2 − 8γ2e−2γℓ + 16γ2(2γℓ − 1)e−4γℓ + O(γ4e−6γℓ),

E2(Qγ

ℓ) = E3(Qγ ℓ) = −2γ2 + 16γ2(2γℓ − 1)e−4γℓ + O(γ4e−6γℓ),

E4(Qγ

ℓ) = −2γ2 + 8γ2e−2γℓ + 16γ2(2γℓ − 1)e−4γℓ + O(γ4e−6γℓ).

15/24

• II. The further Robin eigenvalues EN ⊕+j(Qγ

P)

• First order asymptotics
• Existence of an effective operator for a certain class of curvilinear

polygons

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The further Robin eigenvalues of curvilinear polygons: first order asymptotics

First order asymptotics [Kh., 2018]

If P ⊂ R2 is a curvilinear polygon, then for any fixed j ∈ N, EN ⊕+j(Qγ

P) =

inf specess(T ⊕) γ2 + o(γ2), as γ → +∞, where inf specess(T ⊕) = −1.

16/24

The further Robin eigenvalues of curvilinear polygons: first order asymptotics

First order asymptotics [Kh., 2018]

If P ⊂ R2 is a curvilinear polygon, then for any fixed j ∈ N, EN ⊕+j(Qγ

P) =

inf specess(T ⊕) γ2 + o(γ2), as γ → +∞, where inf specess(T ⊕) = −1.

Can we obtain the next terms ?

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The further Robin eigenvalues of curvilinear polygons: first order asymptotics

First order asymptotics [Kh., 2018]

If P ⊂ R2 is a curvilinear polygon, then for any fixed j ∈ N, EN ⊕+j(Qγ

P) =

inf specess(T ⊕) γ2 + o(γ2), as γ → +∞, where inf specess(T ⊕) = −1.

Can we obtain the next terms ? Examples

• The equilateral triangle [McCartin, 2011] : N ⊕ = 3 and E4(Qγ

T ) = −γ2 + o(1).

• The square (separation of variables): N ⊕ = 4 and for any fixed j ∈ N,

E4+j(Qγ

ℓ) = −γ2 + µj + o(1),

where µj > 0 is the jth element of the disjoint union of the four copies of

• (πk

ℓ )2, k ∈ N

• , i.e.:

π

ℓ

2

,

π

ℓ

2

,

π

ℓ

2

,

π

ℓ

2

, ...,

πk

ℓ

2

,

πk

ℓ

2

,

πk

ℓ

2

,

πk

ℓ

2

, ...

• .

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Non-resonant angles (1/3)

For α ∈ (0, π 2 ) consider the quadrangle Uα,R := OA+

RBRA− R : 2α

R R

x2 x1

Uα

Uα,R

2α x1

O BR A+

R

A−

R

Uα,R

Consider the self-adjoint operator T γ

α,R acting in L2(Uα,R) as:

T γ

α,Ru = −∆u on Uα,R,

∂u ∂ν = γu on ∂Uα,R ∩ ∂Uα, ∂u ∂ν = 0 on ∂Uα,R\∂Uα.

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Non-resonant angles (2/3)

Notation: Nα := {n ∈ N : En(Tα) < −1}.

First eigenvalues of T γ

α,R

As γR → +∞ there holds En(T γ

α,R) = En(Tα)γ2 + O( 1

R2 ), n ∈ {1, ..., Nα}.

2α R R x2 x1 Uα

Uα,R

2α x1

O BR A+

R

A−

R

Uα,R

Localization of the first eigenfunctions

There exist η, C > 0 such that if n ∈ {1, ..., Nα} and un is an eigenfunction of T γ

α,R

associated to En(T γ

α,R) then,

• Uα,R
• 1

γ2 |∇un|2 + |un|2

• eηγ|x|dx ≤ Cun2

L2(Uα,R), as γR → +∞.

The next eigenvalue

ENα+1(T γ

α,R) ≥ −γ2 + o(γ2), as γR → +∞.

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Non-resonant angles (2/3)

Notation: Nα := {n ∈ N : En(Tα) < −1}.

First eigenvalues of T γ

α,R

As γR → +∞ there holds En(T γ

α,R) = En(Tα)γ2 + O( 1

R2 ), n ∈ {1, ..., Nα}.

2α R R x2 x1 Uα

Uα,R

2α x1

O BR A+

R

A−

R

Uα,R

Localization of the first eigenfunctions

There exist η, C > 0 such that if n ∈ {1, ..., Nα} and un is an eigenfunction of T γ

α,R

associated to En(T γ

α,R) then,

• Uα,R
• 1

γ2 |∇un|2 + |un|2

• eηγ|x|dx ≤ Cun2

L2(Uα,R), as γR → +∞.

The next eigenvalue

ENα+1(T γ

α,R) ≥ −γ2 + o(γ2), as γR → +∞.

Remark: The estimates remain true for the Dirichlet version T γ,D

α,R .

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Non-resonant angles (3/3)

Definition

Let γ > 0. The angle α ∈ (0, π

2 ) is said to be non-resonant if there exists C > 0 such

that ENα+1(T γ

α,R) ≥ −γ2 + C

R2 , as R is large.

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Non-resonant angles (3/3)

Definition

Let γ > 0. The angle α ∈ (0, π

2 ) is said to be non-resonant if there exists C > 0 such

that ENα+1(T γ

α,R) ≥ −γ2 + C

R2 , as R is large.

• The definition does not depend on γ :

En(T γ

α,R) = γ2En(T 1 α,γR).

→ Reformulation: ENα+1(T γ

α,R) ≥ −γ2 + C R2 , as γR is large.

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Non-resonant angles (3/3)

Definition

Let γ > 0. The angle α ∈ (0, π

2 ) is said to be non-resonant if there exists C > 0 such

that ENα+1(T γ

α,R) ≥ −γ2 + C

R2 , as R is large.

• The definition does not depend on γ :

En(T γ

α,R) = γ2En(T 1 α,γR).

→ Reformulation: ENα+1(T γ

α,R) ≥ −γ2 + C R2 , as γR is large.

• Dirichlet bracketing: if α is non-resonant then

ENα+1(T γ,D

α,R ) ≥ ENα+1(T γ α,R) ≥ −γ2 + C

R2 , as γR is large.

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Non-resonant angles (3/3)

Definition

Let γ > 0. The angle α ∈ (0, π

2 ) is said to be non-resonant if there exists C > 0 such

that ENα+1(T γ

α,R) ≥ −γ2 + C

R2 , as R is large.

• The definition does not depend on γ :

En(T γ

α,R) = γ2En(T 1 α,γR).

→ Reformulation: ENα+1(T γ

α,R) ≥ −γ2 + C R2 , as γR is large.

• Dirichlet bracketing: if α is non-resonant then

ENα+1(T γ,D

α,R ) ≥ ENα+1(T γ α,R) ≥ −γ2 + C

R2 , as γR is large.

Proposition

Any α ∈ [π 4 , π 2 ) is non-resonant.

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Effective operator for curvilinear polygons: notation

Let P ⊂ R2 be a curvilinear polygon with V ∈ N vertices and such that ∂P = V

i=1 Γi

where each Γi is a smooth curve and: αi := half-aperture of the vertex Ai, ℓi := length of Γi, (0, ℓi) ∋ s → κi(s) the curvature of Γi N ⊕ := V

i=1 Nαi < +∞.

Consider Di := the Dirichlet Laplacian acting in L2(0, ℓi) and Lγ :=

V

• i=1

(Di − γκi) effective operator on ∂P.

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Effective operator for curvilinear polygons

Theorem [Kh.-Ourmi` eres-Bonafos-Pankrashkin, preprint]

Suppose that (H1) for any i ∈ {1, ..., V } the half-angle αi is concave (αi ≥ π

2 ) or αi is

non-resonant, (H2) the curvature κi of each Γi is constant.

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Effective operator for curvilinear polygons

Theorem [Kh.-Ourmi` eres-Bonafos-Pankrashkin, preprint]

Suppose that (H1) for any i ∈ {1, ..., V } the half-angle αi is concave (αi ≥ π

2 ) or αi is

non-resonant, (H2) the curvature κi of each Γi is constant. Then, for any fixed j ∈ N there holds EN ⊕+j(Qγ

P) = −γ2 + Ej(Lγ) + O(1),

γ → +∞, with Lγ := V

i=1 (Di − γκi) .

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Effective operator for curvilinear polygons

Theorem [Kh.-Ourmi` eres-Bonafos-Pankrashkin, preprint]

Suppose that (H1) for any i ∈ {1, ..., V } the half-angle αi is concave (αi ≥ π

2 ) or αi is

non-resonant, (H2) the curvature κi of each Γi is constant. Then, for any fixed j ∈ N there holds EN ⊕+j(Qγ

P) = −γ2 + Ej(Lγ) + O(1),

γ → +∞, with Lγ := V

i=1 (Di − γκi) .

More precisely, denote by κ∗ := maxi κi, then, EN ⊕+j(Qγ

P) = −γ2 − κ∗γ − κ2 ∗

2 + Ej(

• i:κi=κ∗

Di) + O(log γ √γ ), γ → +∞.

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Some words about the proof

• Dirichlet-Neumann bracketing: Pδ := {x ∈ P : dist(x, ∂P) < δ} with

δ → 0 and δγ → +∞ as γ → +∞.

δ δ

≤ ≤

Robin Neumann Robin Dirichlet Robin Pδ Pδ P

Av Av

The choice of δ (non-unique) leads the a priori non-optimal remainder.

• Upper bound: straightforward with the non-resonance condition.

Av Av+1 Av−1

2αv

δ δ

Robin Robin Dirichlet Dirichlet Av

Av−1 Av+1 δ δ

2αv

Dirichlet Dirichlet Dirichlet

• Lower bound: more technical. It is inspired by techniques used in [Post, 2005] for

the study of shrinking waveguides. Important idea: we show that the eigenfunctions associated to EN ⊕+j(Qγ

P ) are

small near the vertices → which appears in the Dirichlet condition at any vertex.

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Existence of resonant angles

If P ⊂ R2 is a polygon with non-resonant angles then: lim

γ→+∞

• EN ⊕+1(Qγ

P ) + γ2

= π2 ℓ2 > 0, where ℓ is the length of the largest side.

The equilateral triangle [McCartin, 2011]

N ⊕ = 3 and E4(Qγ

T ) = −γ2 + o(1), as γ → +∞, i.e. :

lim

γ→+∞

• E4(Qγ

T ) + γ2

= 0. Conclusion : π 6 is resonant.

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The limits of the proof

Let P ⊂ R2 be a general curvilinear polygon. We expect

• the following asymptotics for any fixed j ∈ N,

EN ⊕+j(Qγ

P) = −γ2 + Ej(Lγ) + r(γ), as γ → +∞,

with

• Lγ := − d2

dx2 − γκ on ∂P + boundary conditions called transmission conditions at each vertex depending on the half-apertures αi.

• κ := curvature of ∂Ω,
• r(γ) a ’small’ remainder.

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The limits of the proof

Let P ⊂ R2 be a general curvilinear polygon. We expect

• the following asymptotics for any fixed j ∈ N,

EN ⊕+j(Qγ

P) = −γ2 + Ej(Lγ) + r(γ), as γ → +∞,

with

• Lγ := − d2

dx2 − γκ on ∂P + boundary conditions called transmission conditions at each vertex depending on the half-apertures αi.

• κ := curvature of ∂Ω,
• r(γ) a ’small’ remainder.
• that the associated eigenfunctions are localized near the points of maximal

curvature.

24/24

The limits of the proof

Let P ⊂ R2 be a general curvilinear polygon. We expect

• the following asymptotics for any fixed j ∈ N,

EN ⊕+j(Qγ

P) = −γ2 + Ej(Lγ) + r(γ), as γ → +∞,

with

• Lγ := − d2

dx2 − γκ on ∂P + boundary conditions called transmission conditions at each vertex depending on the half-apertures αi.

• κ := curvature of ∂Ω,
• r(γ) a ’small’ remainder.
• that the associated eigenfunctions are localized near the points of maximal

curvature. → Our method does not directly adapt to the general case if:

• P has resonant angles,
• the maximum of the curvature is reached at a vertex.

24/24

The limits of the proof

Let P ⊂ R2 be a general curvilinear polygon. We expect

• the following asymptotics for any fixed j ∈ N,

EN ⊕+j(Qγ

P) = −γ2 + Ej(Lγ) + r(γ), as γ → +∞,

with

• Lγ := − d2

dx2 − γκ on ∂P + boundary conditions called transmission conditions at each vertex depending on the half-apertures αi.

• κ := curvature of ∂Ω,
• r(γ) a ’small’ remainder.
• that the associated eigenfunctions are localized near the points of maximal

curvature. → Our method does not directly adapt to the general case if:

• P has resonant angles,
• the maximum of the curvature is reached at a vertex.

Thank you !