Mathematical studies of a phase field approach to shape optimization

The goal of this proposal is to develop phase field approximations to study nonlinear inverse problems that can be casted as shape optimization problems. We propose to use a phase field approach since it has several important mathematical and numerical advantages, such as the existence of optimal shapes can be proved rigorously, and in addition both the PDE constraint and the optimality conditions are solved on the same fixed domain throughout the optimization procedure. Furthermore, by sending a small parameter to zero, the optimality conditions from the classical formulation are recovered. Hence, one may view the phase field approach as a consistent approximation for shape optimization problems.