  We prove the global-time existence of weak solutions to the supercooled
Stefan problem. Our result holds in general space dimensions and with a general
class of initial data. In addition, our solution is maximal in the sense of a
certain stochastic order, 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, where the main idea is to introduce a
superharmonic cost function in the optimization problem. We will show that our
choice of the cost function causes 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.
