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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 $p$.

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