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Limits for Monge-Kantorovich mass transport problems

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  • 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 $p$.
    Mathematics Subject Classification: Primary: 35J65, 35J50, 35J55.


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