Limits for Monge-Kantorovich mass transport problems
Jesus Garcia Azorero Juan J. Manfredi I. Peral Julio D. Rossi
Communications on Pure & Applied Analysis 2008, 7(4): 853-865 doi: 10.3934/cpaa.2008.7.853
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 $\partial \Omega$, by performing a suitable extension of the measures to a strip of width $\varepsilon$ near the boundary of the domain $\Omega$ we consider the mass transfer problem for the extensions. Then we study the limit as $\varepsilon$ 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 $p$.
keywords: quasilinear elliptic equations Mass transport Neumann boundary conditions.

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