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Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

Girouard, Alexandre and Karpukhin, Mikhail and Lagacé, Jean (2021) Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems. Geometric and Functional Analysis, 31 (3). pp. 513-561. ISSN 1016-443X. doi:10.1007/s00039-021-00573-5.

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We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is 8πk, which is the best upper bound for the kth area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For k=1, the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

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Alternate Title:Sharp isoperimetric upper bounds for planar Steklov eigenvalues
Additional Information:© 2021 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Received 06 January 2021; Revised 07 June 2021; Accepted 21 June 2021; Published 13 August 2021. The authors would like to thank Iosif Polterovich for introducing them to spectral geometry. This project stemmed from discussions held during the online miniconference on sharp eigenvalue estimates for partial differential operators held by Mark Ashbaugh and Richard Laugesen in lieu of a session at the AMS Sectional Meeting. The authors would like to thank Pier Domenico Lamberti for reading an early version of this manuscript and providing helpful comments, as well as Bruno Colbois for pointing out reference [Ann86]. AG is supported by NSERC and FRQNT. The research of JL was supported by EPSRC grant EP/P024793/1 and the NSERC Postdoctoral Fellowship.
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Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Fonds de recherche du Québec - Nature et technologies (FRQNT)UNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)EP/P024793/1
Issue or Number:3
Record Number:CaltechAUTHORS:20201123-142739007
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Official Citation:Girouard, A., Karpukhin, M. & Lagacé, J. Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems. Geom. Funct. Anal. 31, 513–561 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106794
Deposited By: Tony Diaz
Deposited On:23 Nov 2020 22:34
Last Modified:20 Sep 2021 18:29

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