Existence for the Supercooled Stefan Problem in General Dimensions

In this paper we prove global-time existence of weak solutions to the supercooled Stefan problem in general space dimensions, starting from a general class of initial domain and data. In addition, our solution is maximal, in the sense that its support is maximal among all comparable weak solutions starting from the same initial data. Our approach is based on a free target optimization problem for Brownian stopping times. While this approach is taken previously in [22], the novel feature of our optimization problem here is that it creates particle dynamics with a prescribed initial domain of general geometry. To achieve this, our main idea is to introduce a superharmonic cost function in the optimization problem, which encourages the target measure to accumulate near the prescribed domain boundary as much as possible. A central ingredient in our proof lies in the usage of dual problem: we prove the dual attainment and use the dual optimal solution to characterize the primal optimal solution. It follows in turn that the underlying particle dynamics yields a solution to the supercooled Stefan problem.