On the reformulation of the Classical Stefan problem as a shape optimization problem

This article is concerned with the multidimensional one-phase Stefan problem, which belongs to the class of moving boundary problems. We suggest to reformulate the classical Stefan problem as a shape optimization problem, consisting of an objective functional for the moving boundary and a partial differential equation corresponding to a heat type equation. Minimizing the objective functional subject to the differential equation under consideration is equivalent to solving the Stefan problem. In order to apply gradient-based optimization algorithms, we analytically compute the shape gradient of the objective functional. A numerical example justifies our approach.