  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 [14] and [16] 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 minimizers. 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 its energy norm is not sufficient, but tracking it
in a half an order higher norm will be well-posed. To support our theoretical
results we carry out extensive numerical experiments.
