eBook ISBN:  9781470402426 
Product Code:  MEMO/137/653.E 
List Price:  $46.00 
MAA Member Price:  $41.40 
AMS Member Price:  $27.60 
eBook ISBN:  9781470402426 
Product Code:  MEMO/137/653.E 
List Price:  $46.00 
MAA Member Price:  $41.40 
AMS Member Price:  $27.60 

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 MongeKantorovich problem of optimally rearranging the measure \(\mu{^+}=f^+dx\) onto \(\mu^=f^dy\) can be constructed by studying the \(p\)Laplacian equation \( \mathrm{div}(\vert DU_p\vert^{p2}Du_p)=f^+f^\) 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.

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


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In this volume, the authors demonstrate under some assumptions on \(f^+\), \(f^\) that a solution to the classical MongeKantorovich problem of optimally rearranging the measure \(\mu{^+}=f^+dx\) onto \(\mu^=f^dy\) can be constructed by studying the \(p\)Laplacian equation \( \mathrm{div}(\vert DU_p\vert^{p2}Du_p)=f^+f^\) 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