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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On conformal spectral gap estimates of the Dirichlet-Laplacian


Authors: V. Gol′dshtein, V. Pchelintsev and A. Ukhlov
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 325-335
MSC (2010): Primary 35P15; Secondary 46E35, 30C65
DOI: https://doi.org/10.1090/spmj/1599
Published electronically: February 4, 2020
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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.


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Additional Information

V. Gol′dshtein
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, 8410501, Beer Sheva, Israel
Email: vladimir@math.bgu.ac.il

V. Pchelintsev
Affiliation: Division of Mathematics and Informatics, Tomsk Polytechnic University, Lenin pr. 30, 634050, Tomsk, Russia; and International Laboratory SSP & QF, Tomsk State University, Lenin pr. 36, 634050, Tomsk, Russia
Address at time of publication: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 8410501, Beer Sheva, Israel
Email: vpchelintsev@vtomske.ru

A. Ukhlov
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 8410501, Beer Sheva, Israel
Email: ukhlov@math.bgu.ac.il

DOI: https://doi.org/10.1090/spmj/1599
Keywords: Elliptic equations, Sobolev spaces, conformal mappings
Received by editor(s): October 10, 2018
Published electronically: February 4, 2020
Additional Notes: The first author was supported by the United States–Israel Binational Science Foundation (BSF Grant no. 2014055).
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.
Dedicated: Dedicated to V. Maz’ya on the occasion of his 80th birthday
Article copyright: © Copyright 2020 American Mathematical Society