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 R² with a C² boundary, there exists: = () small enough such that for all 0 < < the maximum of \₁ (- B_{ (x) ): B_ \} is never attained when the ball is close enough to the boundary. In particular it is not obtained when B_ (x) is touching the boundary.