In the optimal partial transport problem, one is interested in transferring a fraction of the mass of one density onto another while minimizing a transportation cost. The free boundary is the object which identifies the region to be transported. In this lecture, we first study the regularity of the free boundary associated to the quadratic cost. Then, we build up a regularity theory for general cost functions. The general principle is that the free boundary is well-behaved away from a potential singular set. We introduce techniques with which to estimate the Hausdorff dimension of the singular set and relate it to classical illumination problems arising in differential geometry.
