The Monge-Ampere Equation

The Monge-Ampere Equation

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Description

Now in its second edition, this monograph explores the Monge-Ampere equation and the latest advances in its study and applications. It provides an essentially self-contained systematic exposition of the theory of weak solutions, including regularity results by L. A. Caffarelli. The geometric aspects of this theory are stressed using techniques from harmonic analysis, such as covering lemmas and set decompositions. An effort is made to present complete proofs of all theorems, and examples and exercises are offered to further illustrate important concepts. Some of the topics considered include generalized solutions, non-divergence equations, cross sections, and convex solutions. New to this edition is a chapter on the linearized Monge-Ampere equation and a chapter on interior Hoelder estimates for second derivatives. Bibliographic notes, updated and expanded from the first edition, are included at the end of every chapter for further reading on Monge-Ampere-type equations and their diverse applications in the areas of differential geometry, the calculus of variations, optimization problems, optimal mass transport, and geometric optics. Both researchers and graduate students working on nonlinear differential equations and their applications will find this to be a useful and concise resource.
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Product details

  • Paperback | 216 pages
  • 155 x 235 x 12.45mm | 361g
  • Basel, Switzerland
  • English
  • Revised
  • Softcover reprint of the original 2nd ed. 2016
  • 3 Illustrations, color; 3 Illustrations, black and white; XIV, 216 p. 6 illus., 3 illus. in color.
  • 3319828061
  • 9783319828060

Back cover copy

Now in its second edition, this monograph explores the Monge-Ampère equation and the latest advances in its study and applications. It provides an essentially self-contained systematic exposition of the theory of weak solutions, including regularity results by L. A. Caffarelli. The geometric aspects of this theory are stressed using techniques from harmonic analysis, such as covering lemmas and set decompositions. An effort is made to present complete proofs of all theorems, and examples and exercises are offered to further illustrate important concepts. Some of the topics considered include generalized solutions, non-divergence equations, cross sections, and convex solutions. New to this edition is a chapter on the linearized Monge-Ampère equation and a chapter on interior Hölder estimates for second derivatives. Bibliographic notes, updated and expanded from the first edition, are included at the end of every chapter for further reading on Monge-Ampère-type equations and their diverse applications in the areas of differential geometry, the calculus of variations, optimization problems, optimal mass transport, and geometric optics. Both researchers and graduate students working on nonlinear differential equations and their applications will find this to be a useful and concise resource.
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Table of contents

Generalized Solutions to Monge-Ampere Equations.- Uniformly Elliptic Equations in Nondivergence Form.- The Cross-sections of Monge-Ampere.- Convex Solutions of detDu=1 in Rn.- Regularity Theory for the Monge-Ampere Equation.- W^2,p Estimates for the Monge-Ampere Equation.- The Linearized Monge-Ampere Equation.- Interior Hoelder Estimates for Second Derivatives.- References.- Index.
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Review Text

"Very clear monograph that will be useful in stimulating new researches on the Monge-Ampère equation, its connections with several research areas and its applications." (Vincenzo Vespri, zbMATH 1356.35004, 2017)
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Review quote

"Very clear monograph that will be useful in stimulating new researches on the Monge-Ampere equation, its connections with several research areas and its applications." (Vincenzo Vespri, zbMATH 1356.35004, 2017)
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About Cristian E. Gutierrez

Cristian Gutierrez is a Professor in the Department of Mathematics at Temple University in Philadelphia, PA, USA. He teaches courses in Partial Differential Equations and Analysis.
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