Girouard, A;
Karpukhin, M;
Lagacé, J;
(2021)
Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems.
Geometric and Functional Analysis
10.1007/s00039-021-00573-5.
(In press).
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Abstract
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.
| Type: | Article |
|---|---|
| Title: | Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00039-021-00573-5 |
| Publisher version: | https://doi.org/10.1007/s00039-021-00573-5 |
| Language: | English |
| Additional information: | 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 http://creativecommons.org/licenses/by/4.0/. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10133811 |
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