On the well-posedness of tracking Dirichlet data for Bernoulli free boundary problems

The aim of this paper is to study the shape optimization method for solving the Bernoulli free boundary problem, a well-known ill-posed problem that seeks the unknown free boundary through Cauchy data. Different formulations have been proposed in the literature that differ in the choice of the objective functional. Specifically, it was shown respectively in Eppler and Harbrecht (SIAM J Control Optim 48:2901–2916, 2010, J Optim Theory Appl 145:17–35, 2010) that tracking Neumann data is well-posed but tracking Dirichlet data is not. In this paper we propose a new well-posed objective functional that tracks Dirichlet data at the free boundary. By calculating the Euler derivative and the shape Hessian of the objective functional we show that the new formulation is well-posed, i.e., the shape Hessian is coercive at the minima. The coercivity of the shape Hessian may ensure the existence of optimal solutions for the nonlinear Ritz–Galerkin approximation method and its convergence, thus is crucial for the formulation. As a summary, we conclude that tracking Dirichlet or Neumann data in their energy norm is not sufficient, but tracking them in a half an order higher norm will be well-posed. To support our theoretical results we carry out extensive numerical experiments.