Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem

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Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem is written by Lawrence C. Evans and published by American Mathematical Soc.. It's available with International Standard Book Number or ISBN identification 0821809385 (ISBN 10) and 9780821809389 (ISBN 13).

In this volume, the authors demonstrate under some assumptions on $f $, $f $ that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure $\mu{ }=f dx$ onto $\mu =f dy$ can be constructed by studying the $p$-Laplacian equation $- \roman{div}(\vert DU_p\vert p-2}Du_p)=f -f $ in the limit as $p\rightarrow\infty$. The idea is to show $u_p\rightarrow u$, where $u$ satisfies $\vert Du\vert\leq 1, -\roman{div}(aDu)=f -f $ for some density $a\geq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f $ and $f $