A note on the maximization of the first Dirichlet eigenvalue for perforated planar domains

TL;DR

This paper proves that for smooth bounded planar domains, the maximum first Dirichlet eigenvalue with a small perforation is not achieved when the perforation touches the boundary, highlighting geometric constraints on eigenvalue optimization.

Contribution

It establishes that the maximum eigenvalue configuration cannot occur with perforations near the boundary for sufficiently small holes, providing new insights into eigenvalue optimization in perforated domains.

Findings

Maximum eigenvalue not attained when perforation touches boundary

Existence of a small epsilon ensuring the maximum is away from boundary

Results hold for domains with $C^2$ boundary

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 $\partial \Omega$.