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

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) 1/24

The Robin eigenvalue problem ν Let Ω ⊂ R d , d ≥ 2, be Lipschitz and bounded. For γ > 0, consider − ∆ u = Eu on Ω , Ω ∂u ∂ν = γu on ∂ Ω . 2/24

The Robin eigenvalue problem ν Let Ω ⊂ R d , d ≥ 2, be Lipschitz and bounded. For γ > 0, consider − ∆ u = Eu on Ω , Ω ∂u ∂ν = γu on ∂ Ω . • Behavior of the eigenvalues E as γ → + ∞ ? 2/24

The Robin eigenvalue problem ν Let Ω ⊂ R d , d ≥ 2, be Lipschitz and bounded. For γ > 0, consider − ∆ u = Eu on Ω , Ω ∂u ∂ν = γu on ∂ Ω . • Behavior of the eigenvalues E as γ → + ∞ ? • Influence of the geometry of Ω on the asymptotics ? 2/24

The Robin eigenvalue problem ν Let Ω ⊂ R d , d ≥ 2, be Lipschitz and bounded. For γ > 0, consider − ∆ u = Eu on Ω , Ω ∂u ∂ν = γu on ∂ Ω . • Behavior of the eigenvalues E as γ → + ∞ ? • Influence of the geometry of Ω on the asymptotics ? Variational approach: the Robin Laplacian Q γ Ω is the unique self-adjoint operator in L 2 (Ω) associated with the closed form � � |∇ u | 2 − γ q γ | u | 2 ds, u ∈ H 1 (Ω) . Ω ( u, u ) = Ω ∂ Ω 2/24

The Robin eigenvalue problem ν Let Ω ⊂ R d , d ≥ 2, be Lipschitz and bounded. For γ > 0, consider − ∆ u = Eu on Ω , Ω ∂u ∂ν = γu on ∂ Ω . • Behavior of the eigenvalues E as γ → + ∞ ? • Influence of the geometry of Ω on the asymptotics ? Variational approach: the Robin Laplacian Q γ Ω is the unique self-adjoint operator in L 2 (Ω) associated with the closed form � � |∇ u | 2 − γ q γ | u | 2 ds, u ∈ H 1 (Ω) . Ω ( u, u ) = Ω ∂ Ω ◮ Let n ∈ N be fixed: E n ( Q γ Ω ) − − − − → γ → + ∞ ? 2/24

Motivation: link with Dirichlet Laplacians acting on domains collapsing on metric graphs (1/2) • The min-max principle gives: E n ( Q γ Ω ) → −∞ as γ → + ∞ , in particular E n ( Q γ Ω ) < 0 for γ large enough . 3/24

Motivation: link with Dirichlet Laplacians acting on domains collapsing on metric graphs (1/2) • The min-max principle gives: E n ( Q γ Ω ) → −∞ as γ → + ∞ , in particular E n ( Q γ Ω ) < 0 for γ large enough . • Estimate on the negative eigenvalues → Dirichlet-Neumann bracketing: Ω ǫ := { x ∈ Ω : dist( x, ∂ Ω) < ǫ } with ǫ := ǫ ( γ ) such that ǫ → 0 as γ → + ∞ . ǫ ǫ Robin Robin Robin ≤ ≤ Ω ǫ Ω ǫ Neumann Dirichlet Ω 3/24

Motivation: link with Dirichlet Laplacians acting on domains collapsing on metric graphs (1/2) • The min-max principle gives: E n ( Q γ Ω ) → −∞ as γ → + ∞ , in particular E n ( Q γ Ω ) < 0 for γ large enough . • Estimate on the negative eigenvalues → Dirichlet-Neumann bracketing: Ω ǫ := { x ∈ Ω : dist( x, ∂ Ω) < ǫ } with ǫ := ǫ ( γ ) such that ǫ → 0 as γ → + ∞ . ǫ ǫ Robin Robin Robin ≤ ≤ Ω ǫ Ω ǫ Neumann Dirichlet Ω • Geometrically the problem share some links with Dirichlet Laplacians acting on domains collapsing on metric graphs [Grieser, 2008; Molchanov-Vainberg, 2007; Post, 2005] Ω ε Γ ε − ∆ u = E n ( ǫ ) u on Ω ǫ , ε → 0 u = 0 on ∂ Ω ǫ . ◮ Let n ∈ N be fixed: E n ( ǫ ) − ǫ → +0 ? − − − → 3/24

Motivation: link with Dirichlet Laplacians acting on domains collapsing on metric graphs (2/2) By rescaling, the limit object Ω 0 := lim ǫ → 0 Ω ǫ is: Ω 0 − ∆ D 0 := Dirichlet Laplacian on Ω 0 , N 0 := # spec disc ( − ∆ D 0 ) < + ∞ , ν := inf spec ess ( − ∆ D 0 ) . Asymptotics of Dirichlet eigenvalues on collapsing domains [Grieser, 2008; Molchanov-Vainberg, 2007; Post, 2005] There exists N ≥ N 0 such that for ǫ → 0 there holds (i) for n ∈ { 1 , ..., N } , E n ( ǫ ) = τ n ǫ 2 + O ( e − c/ǫ ) , τ n ∈ (0 , ν ] , c > 0 . (ii) for any j ∈ N , E N + j ( ǫ ) = ν ǫ 2 + µ j + O ( ǫ ) , where the µ j are the eigenvalues of a quantum graph Laplacian acting on Γ with transmission conditions at the vertices. 4/24

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 C 2 , then for any fixed n ∈ N , Ω ) = − γ 2 + E n ( − ∆ ∂ Ω − γK ) + O (log γ ) , as γ → + ∞ , E n ( Q γ where − ∆ ∂ Ω is the Laplace-Beltrami operator acting in L 2 ( ∂ Ω , ds ) and K denotes the sum of the principal curvatures of ∂ Ω. 5/24

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 C 2 , then for any fixed n ∈ N , Ω ) = − γ 2 + E n ( − ∆ ∂ Ω − γK ) + O (log γ ) , as γ → + ∞ , E n ( Q γ where − ∆ ∂ Ω is the Laplace-Beltrami operator acting in L 2 ( ∂ 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 C 3 then E n ( Q γ Ω ) = − γ 2 − γK max + o ( γ ) where K max := 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 Q γ Ω . 5/24

The corner domains An example : the cube C d ⊂ R d By separation of variables we have C d ) = − dγ 2 + o ( γ 2 ) , as γ → + ∞ . E 1 ( Q γ 6/24

The corner domains An example : the cube C d ⊂ R d By separation of variables we have C d ) = − dγ 2 + o ( γ 2 ) , as γ → + ∞ . E 1 ( Q γ Definition : the corner domains [Dauge, 1988; Grisvard, 1985] Ω ⊂ R d Lipschitz, piecewise smooth, bounded, and for any y ∈ ∂ Ω, there exists V ( y ) : V ( y ) ∼ U y ∩ B (0 , r ) , r > 0 , where U y ⊂ R d is a cone . 0 U x In R 2 : x → corner domains := curvilinear polygons, 0 y 2 α 2 α → tangent cones := infinite sectors. Ω U y 6/24

Asymptotics of the first Robin eigenvalue on corner domains If Ω ⊂ R d is a corner domain we denote by U y the tangent cone of ∂ Ω in y and Λ( U y , γ ) := inf spec( Q γ Λ( U y , γ ) = γ 2 Λ( U y , 1) . U y ) , 7/24

Asymptotics of the first Robin eigenvalue on corner domains If Ω ⊂ R d is a corner domain we denote by U y the tangent cone of ∂ Ω in y and Λ( U y , γ ) := inf spec( Q γ Λ( U y , γ ) = γ 2 Λ( U y , 1) . U y ) , First order asymptotics [Levitin-Parnovski, 2008; Bruneau-Popoff, 2016] If Ω ⊂ R d is a corner domains, then E 1 ( Q γ Ω ) = C Ω γ 2 + o ( γ 2 ) as γ → + ∞ with C Ω := inf y ∈ ∂ Ω Λ( U y , 1) . → The Robin Laplacians on the tangent cones determine the asymptotics of E 1 ( Q γ Ω ): they play the role of model operators . 7/24

Asymptotics of the first Robin eigenvalue on corner domains If Ω ⊂ R d is a corner domain we denote by U y the tangent cone of ∂ Ω in y and Λ( U y , γ ) := inf spec( Q γ Λ( U y , γ ) = γ 2 Λ( U y , 1) . U y ) , First order asymptotics [Levitin-Parnovski, 2008; Bruneau-Popoff, 2016] If Ω ⊂ R d is a corner domains, then E 1 ( Q γ Ω ) = C Ω γ 2 + o ( γ 2 ) as γ → + ∞ with C Ω := inf y ∈ ∂ Ω Λ( U y , 1) . → The Robin Laplacians on the tangent cones determine the asymptotics of E 1 ( Q γ Ω ): they play the role of model operators . → In R 2 , the model operators are the Robin Laplacians acting on infinite sectors. 7/24

I. Study of the model operators : The Robin Laplacians on infinite sectors • Essential spectrum ? Existence of discrete spectrum ? • Properties of the associated eigenfunctions ? 8/24

Study of the model operators (1/2) x 2 Infinite sector of half-aperture α ∈ (0 , π ) : U α U α := { ( x 1 , x 2 ) ∈ R 2 : | arg( x 1 + ix 2 ) | < α } . ν α Robin Laplacian acting on L 2 ( U α ) : O x 1 α T α := Q 1 U α . First properties: [Levitin-Parnovski, 2008; Bruneau-Popoff, 2016; Kh.-Pankrashkin, 2018] For any α ∈ (0 , π ), spec ess ( T α ) = [ − 1 , + ∞ ). • If α ≥ π 2 , spec( T α ) = [ − 1 , + ∞ ). 1 sin 2 α associated to ϕ α ( x ) = exp( − x 1 • If α < π 2 , E 1 ( T α ) = − sin α ). 9/24

Study of the model operators (2/2) Eigenvalue counting function: N α := # { n ∈ N : E n ( T α ) < − 1 } . Finiteness of the discrete spectrum and behavior as α → 0 [Kh.-Pankrashkin, 2018] • N α < + ∞ for any α > 0. • (0 , π 2 ) ∋ α �→ N α is non-increasing. 6 = 1 and hence for any α ∈ [ π 6 , π • N π 2 ), N α = 1. • N α → + ∞ as α → 0. 10/24