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Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem
 
L. C. Evans University of California, Berkeley, Berkeley, CA
W. Gangbo Georgia Institute of Technology, Atlanta, GA
Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem
eBook ISBN:  978-1-4704-0242-6
Product Code:  MEMO/137/653.E
List Price: $46.00
MAA Member Price: $41.40
AMS Member Price: $27.60
Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem
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Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem
L. C. Evans University of California, Berkeley, Berkeley, CA
W. Gangbo Georgia Institute of Technology, Atlanta, GA
eBook ISBN:  978-1-4704-0242-6
Product Code:  MEMO/137/653.E
List Price: $46.00
MAA Member Price: $41.40
AMS Member Price: $27.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1371999; 66 pp
    MSC: Primary 49; 90

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Uniform estimates on the $p$-Laplacian, limits as $p \to \infty $
    • 3. The transport set and transport rays
    • 4. Differentiability and smoothness properties of the potential
    • 5. Generic properties of transport rays
    • 6. Behavior of the transport density along rays
    • 7. Vanishing of the transport density at the ends of rays
    • 8. Approximate mass transfer plans
    • 9. Passage to limits a.e.
    • 10. Optimality
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1371999; 66 pp
MSC: Primary 49; 90

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.

  • Chapters
  • 1. Introduction
  • 2. Uniform estimates on the $p$-Laplacian, limits as $p \to \infty $
  • 3. The transport set and transport rays
  • 4. Differentiability and smoothness properties of the potential
  • 5. Generic properties of transport rays
  • 6. Behavior of the transport density along rays
  • 7. Vanishing of the transport density at the ends of rays
  • 8. Approximate mass transfer plans
  • 9. Passage to limits a.e.
  • 10. Optimality
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.