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

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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title = "On conformal spectral gap estimates of the Dirichlet-Laplacian",

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.",

keywords = "Conformal mappings, Elliptic equations, Sobolev spaces",

author = "V. Gol'dshtein and V. Pchelintsev and A. Ukhlov",

note = "Funding Information: The second author was partially supported by the Ministry of Education and Science of the Russian Federation in the framework of the research Project no. 2.3208.2017/4.6, by RFBR Grant no. 18-31-00011. Funding Information: The first author was supported by the United States–Israel Binational Science Foundation (BSF Grant no. 2014055). Publisher Copyright: {\textcopyright} 2020 American Mathematical Society.",

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doi = "10.1090/spmj/1599",

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AU - Pchelintsev, V.

AU - Ukhlov, A.

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Y1 - 2020/1/1

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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JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 2

ER -

On conformal spectral gap estimates of the Dirichlet-Laplacian

Abstract

Keywords

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