Limits for Monge-Kantorovich mass transport problems

In this paper we study the limit of Monge-Kantorovich mass transfer problems when the involved measures are supported in a small strip near the boundary of a bounded smooth domain, Omega. Given two absolutely continuos measures (with respect to the surface measure) supported on the boundary partial derivative Omega, by performing a suitable extension of the measures to a strip of width epsilon near the boundary of the domain Omega we consider the mass transfer problem for the extensions. Then we study the limit as epsilon goes to zero of the Kantorovich potentials for the extensions and obtain that it coincides with a solution of the original mass transfer problem. Moreover we look for the possible approximations of these problems by solutions to equations involving the p-Laplacian for large values of p.