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 $f-$.
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 $f-$.
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 $f-$.