V. Gol'dshtein, V. Pchelintsev, A. Ukhlov, “On conformal spectral gap estimates of the Dirichlet-Laplacian”, Algebra i Analiz, 2019, Volume 31, Issue 2,Pages <nobr>189

This article is cited in 3 papers

Research Papers

On conformal spectral gap estimates of the Dirichlet-Laplacian

V. Gol'dshtein^a, V. Pchelintsev^bac, A. Ukhlov^a

^a Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 8410501, Beer Sheva, Israel
^b International Laboratory SSP & QF, Tomsk State University, Lenin pr., 36, 634050, Tomsk, Russia
^c Division of Mathematics and Informatics, Tomsk Polytechnic University, Lenin pr., 30, 634050, Tomsk, Russia

Abstract: We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in nonconvex domains $ \Omega \subset \mathbb{R}^2$. 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.

Keywords: elliptic equations, Sobolev spaces, conformal mappings.

MSC: Primary 35P15; Secondary 46E35, 30C65

Received: 10.10.2018

Language: English