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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$43
Individual Members: US$25.80
Institutional Members: US$34.40
Order Code: MEMO/137/653
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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 \(f^-\).

Readership

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

Table of Contents

  • 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
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