A note on the maximization of the first Dirichlet eigenvalue for perforated planar domains
Abstract
In this work we prove that given an open bounded set $\Omega \subset \mathbb{R}^2$ with a $C^2$ boundary, there exists $\epsilon := \epsilon(\Omega)$ small enough such that for all $0 < \delta < \epsilon$ the maximum of $\{\lambda_1(\Omega - B_{\delta}(x)):B_{\delta} \subset \Omega\}$ is never attained when the ball is close enough to the boundary. In particular it is not obtained when $B_\delta(x)$ is touching the boundary $\partial \Omega$.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.01237
- arXiv:
- arXiv:2407.01237
- Bibcode:
- 2024arXiv240701237D
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Optimization and Control;
- Mathematics - Spectral Theory;
- 35J25;
- 49R05;
- 49Q10;
- 35B65;
- 47J10
- E-Print:
- 27 pages, 7 figures. Comments are welcomed