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 $Ω\subset \mathbb{R}^2$ with a $C^2$ boundary, there exists $ε:= ε(Ω)$ small enough such that for all $0 < δ< ε$ the maximum of $\{λ_1(Ω- B_δ(x)):B_δ \subset Ω\}$ is never attained when the ball is close enough to the boundary. In particular it is not obtained when $B_δ(x)$ is touching the boundary $\partial Ω$.