Relaxation approach for optimization of free boundary problems

We consider an optimal control problem (OCP) constrained by a free boundary problem (FBP). FBPs have various applications such as in fluid dynamics, flow in porous media or finance. For this work we study a model FBP given by a Poisson equation in the bulk and a Young‐Laplace equation accounting for surface tension on the free boundary. Transforming this coupled system to a reference domain allows to avoid dealing with shape derivatives. However, this results in highly nonlinear partial differential equation (PDE) coefficients, which makes the OCP rather difficult to handle. Therefore, we present a new relaxation approach by introducing the free boundary as a new control variable, which transforms the original problem into a sequence of simpler optimization problems without free boundary. In this paper, we formally derive the adjoint systems and show numerically that a solution of the original problem can be indeed asymptotically approximated in this way.