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The Monge-Ampère Equation

Author: Cristian E Gutiérrez
Publisher: Boston, MA : Birkhäuser Boston : Imprint : Birkhäuser, 2001.
Series: Progress in nonlinear differential equations and their applications, 44.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Database:WorldCat
Summary:
The classical Monge-Ampère equation has been the center of considerable interest in recent years because of its important role in several areas of applied mathematics. In reflecting these developments, this works stresses the geometric aspects of this beautiful theory, using some techniques from harmonic analysis - covering lemmas and set decompositions. Moreover, Monge-Ampère type equations have applications in  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Cristian E Gutiérrez
ISBN: 9781461201953 1461201950
OCLC Number: 853264338
Description: 1 online resource (xi, 127 pages).
Contents: 1 Generalized Solutions to Monge-Ampere Equations --
1.1 The normal mapping --
1.2 Generalized solutions --
1.3 Viscosity solutions --
1.4 Maximum principles --
1.5 The Dirichlet problem --
1.6 The nonhomogeneous Dirichlet problem --
1.7 Return to viscosity solutions --
1.8 Ellipsoids of minimum volume --
1.9 Notes --
2 Uniformly Elliptic Equations in Nondivergence Form --
2.1 Critical density estimates --
2.2 Estimate of the distribution function of solutions --
2.3 Harnack's inequality --
2.4 Notes --
3 The Cross-sections of Monge-Ampere --
3.1 Introduction --
3.2 Preliminary results --
3.3 Properties of the sections --
3.4 Notes --
4 Convex Solutions of det D2u = 1 in?n --
4.1 Pogorelov's Lemma --
4.2 Interior Hölder estimates of D2u --
4.3 C?estimates of D2u --
4.4 Notes --
5 Regularity Theory for the Monge-Ampère Equation --
5.1 Extremal points --
5.2 A result on extremal points of zeroes of solutions to Monge-Ampère --
5.3 A strict convexity result --
5.4 C1,?regularity --
5.5 Examples --
5.6 Notes --
6 W2pEstimates for the Monge-Ampere Equation --
6.1 Approximation Theorem --
6.2 Tangent paraboloids --
6.3 Density estimates and power decay --
6.4 LP estimates of second derivatives --
6.5 Proof of the Covering Theorem 6.3.3 --
6.6 Regularity of the convex envelope --
6.7 Notes.
Series Title: Progress in nonlinear differential equations and their applications, 44.
Responsibility: by Cristian E. Gutiérrez.
More information:

Abstract:

The classical Monge-Ampère equation has been the center of considerable interest in recent years because of its important role in several areas of applied mathematics. In reflecting these developments, this works stresses the geometric aspects of this beautiful theory, using some techniques from harmonic analysis - covering lemmas and set decompositions. Moreover, 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. The book is an essentially self-contained exposition of the theory of weak solutions, including the regularity results of L.A. Caffarelli. The presentation unfolds systematically from introductory chapters, and an effort is made to present complete proofs of all theorems. Included are examples, illustrations, bibliographical references at the end of each chapter, and a comprehensive index. Topics covered include: * Generalized Solutions * Non-divergence Equations * The Cross-Sections of Monge-Ampère * Convex Solutions of D^2u = 1 in R^n * Regularity Theory * W^2, p Estimates The Monge-Ampère Equation is a concise and useful book for graduate students and researchers in the field of nonlinear equations.

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