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 $\Omega \subset \mathbb{R}^2$ 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 $\partial \Omega$.