  We expose here a novel application of the so-called coupled complex boundary
method -- first put forward by Cheng et al. (2014) to deal with inverse source
problems -- in the framework of shape optimization for solving the exterior
Bernoulli problem, a prototypical model of stationary free boundary problems.
The idea of the method is to transform the overdetermined problem to a complex
boundary value problem with a complex Robin boundary condition coupling the
Dirichlet and Neumann boundary conditions on the free boundary. Then, we
optimize the cost function constructed by the imaginary part of the solution in
the whole domain in order to identify the free boundary. We also prove the
existence of the shape derivative of the complex state with respect to the
domain. Afterwards, we compute the shape gradient of the cost functional, and
characterize its shape Hessian at the optimal domain under a strong, and then a
mild regularity assumption on the domain. We then examine the instability of
the proposed method by proving the compactness of the latter expression. Also,
we devise an iterative algorithm based on a Sobolev gradient scheme via finite
element method to solve the minimization problem. Finally, we illustrate the
applicability of the method through several numerical examples, both in two and
three spatial dimensions.
