Regularity Properties of Optimal Maps Between Nonconvex Domains in the Plane

Given two bounded open subsets , < subset of>d, and two densities f and g concentrated on and , respectively, we investigate the regularity of the optimal map delta phi sending f onto g. We show that, if f and g are both bounded away from zero and infinity, then we can find two open sets '< subset of> and '< subset of> such that f and g are concentrated on ' and ' respectively, and delta phi: '' is a homeomorphism. Moreover, if f and g are smooth, then delta phi is a smooth diffeomorphism between ' and '. Finally, we give a quite precise description of the singular set of phi, showing that it is a 1-dimensional manifold of class C1 out of a countable set.