A Perturbation Argument for a Monge-AmpSre Type Equation Arising in Optimal Transportation

We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge-AmpSre equation in order to obtain, first, interior C-1,C-1 estimates for the potential and, second, interior Holder estimates for second derivatives. In particular, we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary.