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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
Available Formats:
Electronic 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 Click above image for expanded view 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 Available Formats:  Electronic 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 $f^-$.

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
• Requests

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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 $f^-$.

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 reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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