Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem
Author： L. C. Evans;W. Gangbo
Publisher： American Mathematical Society
Publication year： 2013
E-ISBN: 9781470402426
P-ISBN(Paperback): 9780821809389
P-ISBN(Hardback): 9780821809389
Subject： O221 On mathematics planning (planning)
Keyword： Differential Equations
Language： ENG
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Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem
Description
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 $- \mathrm{div}(\vert DU_p\vert^{p-2}Du_p)=f^+-f^-$ in the limit as $p\rightarrow\infty$. The idea is to show $u_p\rightarrow u$, where $u$ satisfies $\vert Du\vert\leq 1,-\mathrm{div}(aDu)=f^+-f^-$ for some density $a\geq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f^+$ and $f^-$.