| 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 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 137; 1999; 66 ppMSC: 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 \(f^-\).
ReadershipGraduate students and research mathematicians working in optimal control problems involving ODEs.
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Table of Contents
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Chapters
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1. Introduction
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2. Uniform estimates on the $p$-Laplacian, limits as $p \to \infty $
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3. The transport set and transport rays
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4. Differentiability and smoothness properties of the potential
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5. Generic properties of transport rays
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6. Behavior of the transport density along rays
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7. Vanishing of the transport density at the ends of rays
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8. Approximate mass transfer plans
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9. Passage to limits a.e.
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10. Optimality
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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 \(f^-\).
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
