  We 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. PDEs and a free boundary
problem approach are used to address the problem in general dimensions with
space-time inhomogeneous costs given by Lagrangian integrals along the paths.
We introduce an Eulerian---mass flow---formulation of the problem, whose 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.
