**Memoirs of the American Mathematical Society**

1999;
66 pp;
Softcover

MSC: Primary 49; 90;

Print ISBN: 978-0-8218-0938-9

Product Code: MEMO/137/653

List Price: $46.00

AMS Member Price: $27.60

MAA member Price: $41.40

**Electronic ISBN: 978-1-4704-0242-6
Product Code: MEMO/137/653.E**

List Price: $46.00

AMS Member Price: $27.60

MAA member Price: $41.40

# Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem

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*L. C. Evans; W. Gangbo*

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

# Table of Contents

## Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem

- Contents vii8 free
- 1. Introduction 110 free
- 2. Uniform estimates on the p-Laplacian, limits as p [omitted] ∞ 716 free
- 3. The transport set and transport rays 1625
- 4. Differentiability and smoothness properties of the potential 2231
- 5. Generic properties of transport rays 2837
- 6. Behavior of the transport density along rays 3645
- 7. Vanishing of the transport density at the ends of rays 4453
- 8. Approximate mass transfer plans 5261
- 9. Passage to limits a.e. 5665
- 10. Optimality 6271
- Appendix: Approximating semiconcave and semiconvex functions by C[sup(2)] functions 6372
- Bibliography 6574