I will consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. A PDE (free boundary problem) approach is used to address the problem in general dimensions with space-time inhomogeneous costs given by Lagrangian integrals along the paths. An Eulerian---mass flow---formulation of the problem is introduced. Its dual is given by Hamilton-Jacobi-Bellman type variational inequalities. Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space-time. This complements and provides a constructive PDE alternative to recent results of Beiglb\"ock, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.
