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