On a Kohn-Vogelius like formulation of free boundary problems

The present paper is concerned with the solution of a Bernoulli type free boundary problem by means of shape optimization. Two state functions are introduced, namely one which satisfies the mixed boundary value problem, whereas the second one satisfies the pure Dirichlet problem. The shape problem under consideration is the minimization of the L ²-distance of the gradients of the state functions. We compute the corresponding shape gradient and Hessian. By the investigation of sufficient second order conditions we prove algebraic ill-posedness of the present formulation. Our theoretical findings are supported by numerical experiments.