On conformal spectral gap estimates of the Dirichlet-Laplacian

V. Gol'dshtein; V. Pchelintsev; A. Ukhlov

doi:10.1090/spmj/1599

On conformal spectral gap estimates of the Dirichlet-Laplacian

V. Gol'dshtein, V. Pchelintsev, A. Ukhlov

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains Ω ⊂ ℝ². With the help of these estimates, we obtain asymptotically sharp inequalities of ratios of eigenvalues in the framework of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains in terms of conformal (hyperbolic) geometry.

Original language	English
Pages (from-to)	325-335
Number of pages	11
Journal	St. Petersburg Mathematical Journal
Volume	31
Issue number	2
DOIs	https://doi.org/10.1090/spmj/1599
State	Published - 1 Jan 2020

Keywords

Conformal mappings
Elliptic equations
Sobolev spaces

ASJC Scopus subject areas

Analysis
Algebra and Number Theory
Applied Mathematics

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10.1090/spmj/1599

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abstract = "We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains Ω ⊂ ℝ2. With the help of these estimates, we obtain asymptotically sharp inequalities of ratios of eigenvalues in the framework of the Payne-P{\'o}lya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains in terms of conformal (hyperbolic) geometry.",

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AU - Ukhlov, A.

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N2 - We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains Ω ⊂ ℝ2. With the help of these estimates, we obtain asymptotically sharp inequalities of ratios of eigenvalues in the framework of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains in terms of conformal (hyperbolic) geometry.

AB - We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains Ω ⊂ ℝ2. With the help of these estimates, we obtain asymptotically sharp inequalities of ratios of eigenvalues in the framework of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains in terms of conformal (hyperbolic) geometry.

KW - Conformal mappings

KW - Elliptic equations

KW - Sobolev spaces

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On conformal spectral gap estimates of the Dirichlet-Laplacian

Abstract

Keywords

ASJC Scopus subject areas

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