Effective operators for Robin eigenvalues in domains with corners

Khalile, Magda; Ourmières-Bonafos, Thomas; Pankrashkin, Konstantin

doi:10.5802/aif.3400

Cited by 7 publications

(4 citation statements)

References 41 publications

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“…in the sector 𝔖 𝛼 , cf. [24,25]. We are interested in solutions of (3.3) which approximately behave as a combination of an incoming and an outgoing plane wave, that is, as This notation is chosen to match [33].…”

Section: Plane Wave Solutions In a Sectormentioning

confidence: 99%

“…Consider the Robin boundary value problem

\begin{equation} \Delta \Phi = 0\quad \text{ in }\mathfrak {S}_{\alpha },\qquad \frac{\partial \Phi }{\partial n}=\Phi \quad \text{ on }\partial \mathfrak {S}_{\alpha } \end{equation}

in the sector

\mathfrak {S}_{\alpha }

, cf. [24, 25]. We are interested in solutions of () which approximately behave as a combination of an incoming and an outgoing plane wave, that is, as

\begin{equation} \Phi (z)= \Phi _{\alpha }^{(\mathbf {h}_{{\rm in}},\mathbf {h}_{{\rm out}})}(z):=W^{\mathbf {h}_{{\rm out}}}_{{\rm out},\alpha }(z)+W^{\mathbf {h}_{{\rm in}}}_{{\rm in},\alpha }(z)+R^{\mathbf {h}_{{\rm in}},\mathbf {h}_{{\rm out}}}_{\alpha }(z), \end{equation}

with some vectors

\mathbf {h}_{{\rm in}}

and

\mathbf {h}_{{\rm out}}\in \mathbb {C}^2

, where the remainder

R=R^{\mathbf {h}_{{\rm in}},\mathbf {h}_{{\rm out}}}_{\alpha }(z)

…”

Section: Auxiliary Problems In a Sector: Peters Solutionsmentioning

confidence: 99%

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Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

Levitan

Parnovski

Polterovich

et al. 2022

Proceedings of London Math Soc

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We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunctions. Our proofs are based on an explicit construction of quasimodes. We use a variety of methods, including ideas from spectral geometry, layer potential analysis, and some new techniques tailored to our problem.M S C 2 0 2 0 35P20 (primary), 34B45 (secondary) Contents

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Section: Plane Wave Solutions In a Sectormentioning

confidence: 99%

“…Consider the Robin boundary value problem

\begin{equation} \Delta \Phi = 0\quad \text{ in }\mathfrak {S}_{\alpha },\qquad \frac{\partial \Phi }{\partial n}=\Phi \quad \text{ on }\partial \mathfrak {S}_{\alpha } \end{equation}

in the sector

\mathfrak {S}_{\alpha }

, cf. [24, 25]. We are interested in solutions of () which approximately behave as a combination of an incoming and an outgoing plane wave, that is, as

\begin{equation} \Phi (z)= \Phi _{\alpha }^{(\mathbf {h}_{{\rm in}},\mathbf {h}_{{\rm out}})}(z):=W^{\mathbf {h}_{{\rm out}}}_{{\rm out},\alpha }(z)+W^{\mathbf {h}_{{\rm in}}}_{{\rm in},\alpha }(z)+R^{\mathbf {h}_{{\rm in}},\mathbf {h}_{{\rm out}}}_{\alpha }(z), \end{equation}

with some vectors

\mathbf {h}_{{\rm in}}

and

\mathbf {h}_{{\rm out}}\in \mathbb {C}^2

, where the remainder

R=R^{\mathbf {h}_{{\rm in}},\mathbf {h}_{{\rm out}}}_{\alpha }(z)

…”

Section: Auxiliary Problems In a Sector: Peters Solutionsmentioning

confidence: 99%

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

Levitan

Parnovski

Polterovich

et al. 2022

Proceedings of London Math Soc

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show abstract

“…The case of less regularity, namely when Ω has a finite number of "model corners" and the asymptotic behaviour is different, has also been extensively considered [17,41,42,43,50]. We refer to [18] for a recent summary of the problem, its history and more references.…”

Section: Introductionmentioning

confidence: 99%

On the eigenvalues of the Robin Laplacian with a complex parameter

2022

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We study the spectrum of the Robin Laplacian with a complex Robin parameter α on a bounded Lipschitz domain Ω. We start by establishing a number of properties of the corresponding operator, such as generation properties, analytic dependence of the eigenvalues and eigenspaces on α ∈ C, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of α: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case α ∈ R. For the asymptotics of the eigenvalues as α → ∞ in C, in place of the min-max characterisation of the eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the real case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to show that along every analytic curve of eigenvalues, the Robin eigenvalues either diverge absolutely in C or converge to the Dirichlet spectrum, as well as to classify all possible points of accumulation of Robin eigenvalues for large α. We also give a comprehensive treatment of the special cases where Ω is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension d ≥ 2 all eigenvalues converge to the Dirichlet spectrum if Re α remains bounded from below as α → ∞, while if Re α → −∞, then there is an infinite family of divergent eigenvalue curves, each of which behaves asymptotically like −α 2 .

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“…For a review of spectral problems with Robin boundary conditions we refer to [3]. In particular, the eigenvalues of Robin Laplacians on infinite cones play an important role in the strong coupling asymptotics of Robin eigenvalues on general domains as discussed in [2,7,10]. In addition, such operators attract some attention as examples of geometric "long-range" configurations producing an infinite discrete spectrum [1,4,12].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of Robin eigenvalues on sharp infinite cones

Pankrashkin¹,

Vogel²

2022

Preprint

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For ε > 0 and n ∈ N consider the infinite coneand the operator Q α ε acting as the Laplacian u → −∆u on Ω ε with the Robin boundary condition ∂ ν u = αu at ∂Ω ε , where ∂ ν is the outward normal derivative and α > 0. It is known from numerous earlier works that the essential spectrum of Q α ε is [−α 2 , +∞) and that the discrete spectrum is finite for n = 1 and infinite for n ≥ 2, but the behavior of individual eigenvalues with respect to the geometric parameter ε was only addressed for n = 1 so far. In the present work we consider arbitrary n ≥ 2 and look at the spectral asymptotics as ε becomes small, i.e. as the cone becomes "sharp" and collapses to its central axis. Our main result is as follows: if n ≥ 2, α > 0 and j ∈ N are fixed, then the jth eigenvalue E j (Q α ε ) of Q α ε behaves as

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Effective operators for Robin eigenvalues in domains with corners

Abstract: Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org

Cited by 7 publications

References 41 publications

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

On the eigenvalues of the Robin Laplacian with a complex parameter

Asymptotics of Robin eigenvalues on sharp infinite cones

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