Ebook: Differential Equations Methods for the Monge-Kantorevich Mass Transfer Problem

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 $- mathrm{div}(vert DU_pvert^{p-2}Du_p)=f^+-f^-$ in the limit as $prightarrowinfty$. The idea is to show $u_prightarrow u$, where $u$ satisfies $vert Duvertleq 1,-mathrm{div}(aDu)=f^+-f^-$ for some density $ageq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f^+$ and $f^-$.