Differential Equations Methods for the Monge-Kantorevich Mass Transfer Problem (Memoirs of the American Mathematical Society)

By Wilfrid Gangbo and L.C. Evans

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 $f^-$.

• Format: Paperback
• Language: English
• ISBN: 0821809385
• ISBN13: 9780821809389
• Release Date: January 1999
• Publisher: Amer Mathematical Society
• Length: 66 Pages
• Weight: 0.35 lbs.
• Dimensions: 10.0" x 0.5" x 7.0"