Sufficient condition for the existence of solutions of a free boundary problem

In this paper we study the existence of critical points of a functional depending on $\Omega \subset \mathbb {R}^{2}$ through its perimeter and the solution of the Dirichlet problem in $\Omega$, under the constraint that the measure of $\Omega$ is given. We give a sufficient condition for the existence of critical points using the implicit function theorem.