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 ⊂ R² with a C² boundary, there exists ε := ε(Ω) small enough such that for all 0 < δ < ε the maximum of \λ₁(Ω - Bδ(x)):Bδ ⊂ 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 ∂ Ω.