The Steklov spectrum of surfaces: asymptotics and invariants

Girouard, Alexandre; Parnovski, Leonid; Polterovich, Iosif; Sher, David A.

doi:10.1017/s030500411400036x

Cited by 31 publications

(67 citation statements)

References 19 publications

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“…Moreover, λ ∂Ω 2j = λ ∂Ω 2j+1 + O(j −∞ ), for j ≥ 1. For more general smooth Riemann surfaces a similar result was established in [19]. In the following we present some spectra for some explicit domains, starting with the unit circle.…”

Section: Figuresupporting

confidence: 59%

Optimization of Steklov-Neumann eigenvalues

Ammari

Imeri

Nigam

2020

Journal of Computational Physics

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This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the surface. The main objective is to determine the right extent of the covering pieces, so that any shock inside the container yields a resonance. To this end, an algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces. Proofs for these formulas are established. Furthermore, this paper displays some results concerning bounds and examples with regards to the governing problem. Mathematics Subject Classification (MSC2000). 35R30, 35C20.Keywords. Steklov eigenvalue problem, boundary integral operators, mixed boundary conditions, perturbations formulas.

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Section: Figuresupporting

confidence: 59%

Optimization of Steklov-Neumann eigenvalues

Ammari

Imeri

Nigam

2020

Journal of Computational Physics

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“…It is hoped that using the vast amount of literature on the Dirichlet-to-Neumann map, (see, e.g., [43][44][45][46][47][48][49]), new insights can be gained on what the decisive topological and geometrical factors are that lead to attractive or repulsive Casimir forces.…”

Section: Discussionmentioning

confidence: 99%

Gluing Formula for Casimir Energies

Kirsten

Lee

2018

Symmetry

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Abstract:We provide a completely new perspective for the analysis of Casimir forces in very general piston configurations. To this end, in order to be self-contained, we prove a "gluing formula" well known in mathematics and relate it with Casimir forces in piston configurations. At the center of our description is the Dirichlet-to-Neumann operator, which encodes all the information about those forces. As an application, the results for previously considered piston configurations are reproduced in a streamlined fashion.

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“…If Γ is regular, it is sufficient to apply a conform map to project Γ to a sphere and, hence, to obtain the same result (for the conformal map technics see [18] the proof of Theorem 1.4, but also [17] and [8]). For the general case of a d -set Γ , it is more natural to use given in the previous Section definitions of the Dirichlet-to-Neumann operators.…”

Section: Proof Of Theorem 12 and Final Remarksmentioning

confidence: 99%

“…From [9], we also have that Ker A = {0} , since 0 is the eigenvalue of the Neumann eigenvalue problem for the Laplacian. For the Weil asymptotic formulas for the distribution of the eigenvalues of the Dirichlet-to-Neumann operator there are results for bounded smooth compact Riemannian manifolds with C ∞ boundaries [18], for polygons [19] and more general class of plane domains [17] and also for a bounded domain with a fractal boundary [39].…”

Section: Spectral Properties Of the Poincaré-steklov Operator Definedmentioning

confidence: 99%

Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by <i>d</i>-sets

Arfi

Rozanova-Pierrat

2019

Discrete &Amp; Continuous Dynamical Systems - S

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In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in R n , we generalize the definition of the Poincaré-Steklov operator to d -set boundaries, n − 2 < d < n , and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of n -sets. The results are obtained thanks to a generalization of the continuity and compactness properties of the trace and extension operators in Sobolev, Lebesgue and Besov spaces, in particular, by a generalization of the classical Rellich-Kondrachov Theorem of compact embeddings for n and d -sets.

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The Steklov spectrum of surfaces: asymptotics and invariants

Cited by 31 publications

References 19 publications

Optimization of Steklov-Neumann eigenvalues

Optimization of Steklov-Neumann eigenvalues

Gluing Formula for Casimir Energies

Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by <i>d</i>-sets

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