- Covers the latest advances in the study of the Monge-Ampère equation and its applications
- Includes new chapters on the Harnack inequality for the linearized Monge-Ampère equation and on interior Hölder estimates for second derivatives
- Bibliographic notes provided at the end of each chapter for further exploration of topics
Buy this book
- About this book
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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.
- About the authors
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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.
- Reviews
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“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)
- Table of contents (8 chapters)
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Generalized Solutions to Monge–Ampère Equations
Pages 1-39
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Uniformly Elliptic Equations in Nondivergence Form
Pages 41-54
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The Cross-Sections of Monge–Ampère
Pages 55-76
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Convex Solutions of det D 2 u = 1 in ℝ n
Pages 77-89
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Regularity Theory for the Monge–Ampère Equation
Pages 91-122
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Table of contents (8 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- The Monge-Ampère Equation
- Authors
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- Cristian Gutierrez
- Series Title
- Progress in Nonlinear Differential Equations and Their Applications
- Series Volume
- 89
- Copyright
- 2016
- Publisher
- Birkhäuser Basel
- Copyright Holder
- Springer International Publishing
- eBook ISBN
- 978-3-319-43374-5
- DOI
- 10.1007/978-3-319-43374-5
- Hardcover ISBN
- 978-3-319-43372-1
- Softcover ISBN
- 978-3-319-82806-0
- Series ISSN
- 1421-1750
- Edition Number
- 2
- Number of Pages
- XIV, 216
- Number of Illustrations
- 3 b/w illustrations, 3 illustrations in colour
- Topics
