# American Institute of Mathematical Sciences

2008, 7(4): 853-865. doi: 10.3934/cpaa.2008.7.853

## Limits for Monge-Kantorovich mass transport problems

 1 Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid, Spain 2 Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States 3 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid 4 IMDEA Matematicas, C-IX, Campus UAM, 28049 Madrid, Spain

Received  September 2006 Revised  February 2007 Published  April 2008

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 $p$.
Citation: Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 853-865. doi: 10.3934/cpaa.2008.7.853
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