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The Monge-Ampère Equation and Its Applications

2017 Differential equations, Partial
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Release : 2017
Genre : Differential equations, Partial
Kind : eBook
Book Rating : 705/5 ( reviews)

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Book Synopsis The Monge-Ampère Equation and Its Applications by : Alessio Figalli

Download or read book The Monge-Ampère Equation and Its Applications written by Alessio Figalli. This book was released on 2017. Available in PDF, EPUB and Kindle. Book excerpt: The Monge-Ampere equation is one of the most important partial differential equations, appearing in many problems in analysis and geometry. This monograph is a comprehensive introduction to the existence and regularity theory of the Monge-Ampere equation and some selected applications; the main goal is to provide the reader with a wealth of results and techniques he or she can draw from to understand current research related to this beautiful equation. The presentation is essentially self-contained, with an appendix that contains precise statements of all the results used from different areas (linear algebra, convex geometry, measure theory, nonlinear analysis, and PDEs). This book is intended for graduate students and researchers interested in nonlinear PDEs: explanatory figures, detailed proofs, and heuristic arguments make this book suitable for self-study and also as a reference.

The Monge—Ampère Equation

2001-05-11 Mathematics
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Release : 2001-05-11
Genre : Mathematics
Kind : eBook
Book Rating : 771/5 ( reviews)

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Book Synopsis The Monge—Ampère Equation by : Cristian E. Gutierrez

Download or read book The Monge—Ampère Equation written by Cristian E. Gutierrez. This book was released on 2001-05-11. Available in PDF, EPUB and Kindle. Book excerpt: The Monge-Ampère equation has attracted considerable interest in recent years because of its important role in several areas of applied mathematics. Monge-Ampère type equations have applications in the areas of differential geometry, the calculus of variations, and several optimization problems, such as the Monge-Kantorovitch mass transfer problem. This book stresses the geometric aspects of this beautiful theory, using techniques from harmonic analysis – covering lemmas and set decompositions.

The Monge—Ampère Equation

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 950/5 ( reviews)

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Book Synopsis The Monge—Ampère Equation by : Cristian E. Gutierrez

Download or read book The Monge—Ampère Equation written by Cristian E. Gutierrez. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: The Monge-Ampère equation has attracted considerable interest in recent years because of its important role in several areas of applied mathematics. Monge-Ampère type equations have applications in the areas of differential geometry, the calculus of variations, and several optimization problems, such as the Monge-Kantorovitch mass transfer problem. This book stresses the geometric aspects of this beautiful theory, using techniques from harmonic analysis – covering lemmas and set decompositions.

The Monge-Ampère Equation

2016-10-22 Mathematics
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Release : 2016-10-22
Genre : Mathematics
Kind : eBook
Book Rating : 741/5 ( reviews)

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Book Synopsis The Monge-Ampère Equation by : Cristian E. Gutiérrez

Download or read book The Monge-Ampère Equation written by Cristian E. Gutiérrez. This book was released on 2016-10-22. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Analysis of Monge–Ampère Equations

2024-03-07 Mathematics
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Release : 2024-03-07
Genre : Mathematics
Kind : eBook
Book Rating : 204/5 ( reviews)

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Book Synopsis Analysis of Monge–Ampère Equations by : Nam Q. Le

Download or read book Analysis of Monge–Ampère Equations written by Nam Q. Le. This book was released on 2024-03-07. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a systematic analysis of the Monge–Ampère equation, the linearized Monge–Ampère equation, and their applications, with emphasis on both interior and boundary theories. Starting from scratch, it gives an extensive survey of fundamental results, essential techniques, and intriguing phenomena in the solvability, geometry, and regularity of Monge–Ampère equations. It describes in depth diverse applications arising in geometry, fluid mechanics, meteorology, economics, and the calculus of variations. The modern treatment of boundary behaviors of solutions to Monge–Ampère equations, a very important topic of the theory, is thoroughly discussed. The book synthesizes many important recent advances, including Savin's boundary localization theorem, spectral theory, and interior and boundary regularity in Sobolev and Hölder spaces with optimal assumptions. It highlights geometric aspects of the theory and connections with adjacent research areas. This self-contained book provides the necessary background and techniques in convex geometry, real analysis, and partial differential equations, presents detailed proofs of all theorems, explains subtle constructions, and includes well over a hundred exercises. It can serve as an accessible text for graduate students as well as researchers interested in this subject.

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