Abstract: We consider an optimal transport problem where the cost depends on the stopping time of Brownian motion from a given distribution to another. When the target measure is fixed, it is often called the optimal Skorokhod embedding problem in the literature, a popular topic in math finance. Under a monotonicity assumption on the cost, the optimal stopping time is given by the hitting time to a space-time barrier set. When the target measure is optimized under an upper bound constraint, we will show that the optimal barrier set leads us to the Stefan problem, a free boundary problem for the heat equation describing phase transition between water and ice.  Our approach is flexible and can be applied for instance to the study of nonlocal Stefan problem, which corresponds to stopping times of alpha-stable process instead of Brownian motion. Joint works with Young-Heon Kim (UBC), Raymond Chu (UCLA), and Kyeong-Sik Nam (KAIST).