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

In this work we prove that given an open bounded set $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with a $C^2$ boundary, there exists $\epsilon :=\epsilon (\mathrm{\Omega })$ small enough such that for all $0<\delta <\epsilon$ the maximum of $\{{\lambda }_1(\mathrm{\Omega }-B_{\delta }(x)):B_{\delta }\subset \mathrm{\Omega }\}$ 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 $\mathrm{\partial }\mathrm{\Omega }$.