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We formulate and solve a free target optimal Brownian stopping problem from agiven distribution while the target distribution is free and is conditioned tosatisfy a given density height constraint. The free target optimization problemexhibits monotonicity, from which a remarkable universality follows, in thesense that the optimal target is independent of its Lagrangian cost type. Inparticular, the solutions to this optimization problem generate solutions toboth unstable and stable type of the Stefan problem, where former stands forfreezing of supercooled fluid $(St_1)$ and the latter for ice melting $(St_2)$.This unified approach to both types of Stefan problem is new. In particular weobtain global-time existence and weak-strong uniqueness for the ill-posedfreezing problem $(St_1)$, for a given initial data and for a well-preparedclass of initial domains generated from the initial data.

The Stefan problem and free targets of optimal Brownian martingale transport