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 $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ with a $C^2$ boundary, there exists $\hat{I}\mu :=\hat{I}\mu (\hat{I}\text{c◯})$ small enough such that for all $0<\hat{I}´<\hat{I}\mu$ the maximum of $\{\hat{I}{»}_1(\hat{I}\text{c◯}-B_{\hat{I}}´(x)):B_{\hat{I}}´\subset \hat{I}\text{c◯}\}$ is never attained when the ball is close enough to the boundary. In particular it is not obtained when $B_{\hat{I}}´(x)$ is touching the boundary $\partial \hat{I}\text{c◯}$.

27 pages, 7 figures. Comments are welcomed