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Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem
L. C. Evans, University of California, Berkeley, CA, and W. Gangbo, Georgia Institute of Technology, Atlanta, GA
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Memoirs of the American Mathematical Society
1999; 66 pp; softcover
Volume: 137
ISBN-10: 0-8218-0938-5
ISBN-13: 978-0-8218-0938-9
List Price: US$46 Individual Members: US$27.60
Institutional Members: US\$36.80
Order Code: MEMO/137/653

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 $$f^-$$.

Graduate students and research mathematicians working in optimal control problems involving ODEs.

• Introduction
• Uniform estimates on the $$p$$-Laplacian, limits as $$p\to\infty$$
• The transport set and transport rays
• Differentiability and smoothness properties of the potential
• Generic properties of transport rays
• Behavior of the transport density along rays
• Vanishing of the transport density at the ends of rays
• Approximate mass transfer plans
• Passage to limits a.e.
• Optimality
• Appendix: Approximating semiconcave and semiconvex functions by $$C^2$$ functions
• Bibliography