Existence and Symmetry results for some overdetermined free boundary problems

In this paper, we prove that a domain which verifies some integral inequality is either (strictly) contained in the solution of some free boundary problem, or it coincides with an $N$-ball. We also present new overdetermined value problems which have an $N$-ball as a solution. To reach our results, we use an integral identity which involves the domain derivative of the solution of Dirichlet problem.