  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 $\partial \Omega$.
