Analysis of Monge-Ampère equations

• Verfasst von: Le, Nam Q. [VerfasserIn]
• Titel: Analysis of Monge-Ampère equations
• Verf.angabe: Nam Q. Le
• Verlagsort: Providence, Rhode Island
• Verlag: American Mathematical Society
• E-Jahr: 2024
• Jahr: [2024]
• Umfang: 1 Online-Ressource (xx, 576 Seiten)
• Gesamttitel/Reihe: Graduate studies in mathematics ; 240
• Fussnoten: Description based on publisher supplied metadata and other sources
• ISBN: 978-1-4704-7624-3
• Schlagwörter: (s)Monge-Ampère-Differentialgleichung / (s)Linearisierung / (s)Monge-Ampère-Maß / (s)Regularität / (s)Randverhalten / (s)Konvexität / (s)Viskositätslösung / (s)Spektraltheorie / (s)Harnack-Ungleichung
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe: Le, Nam Q.: Analysis of Monge-Ampère equations. - Providence, Rhode Island : American Mathematical Society, 2024. - xx, 576 Seiten
• Sach-SW: Partial differential equations
• Sach-SW: Convex and discrete geometry
• Sach-SW: Calculus of variations and optimal control; optimization
• K10plus-PPN: 1881663957
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

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.

Abstract

Cover -- Title page -- Contents -- Preface -- Notation -- Chapter 1. Introduction -- 1.1. The Monge-Ampère Equation -- 1.2. The Linearized Monge-Ampère Equation -- 1.3. Plan of the Book -- 1.4. Notes -- 1.5. Problems -- Chapter 2. Geometric and Analytic Preliminaries -- 2.1. Convex Sets -- 2.2. The Hausdorff Distance -- 2.3. Convex Functions and the Normal Mapping -- 2.4. Boundary Principal Curvatures and Uniform Convexity -- 2.5. Calculus with Determinant -- 2.6. John's Lemma -- 2.7. Review of Measure Theory and Functional Analysis -- 2.8. Review of Classical PDE Theory -- 2.9. Pointwise Estimates and Perturbation Argument -- 2.10. Problems -- 2.11. Notes -- Part 1. The Monge-Ampère Equation -- Chapter 3. Aleksandrov Solutions and Maximum Principles -- 3.1. Motivations and Heuristics -- 3.2. The Monge-Ampère Measure and Aleksandrov Solutions -- 3.3. Maximum Principles -- 3.4. Global Hölder Estimates and Compactness -- 3.5. Comparison Principle and Global Lipschitz Estimates -- 3.6. Explicit Solutions -- 3.7. The Dirichlet Problem and Perron's Method -- 3.8. Comparison Principle with Nonconvex Functions -- 3.9. Problems -- 3.10. Notes -- Chapter 4. Classical Solutions -- 4.1. Special Subdifferential at the Boundary -- 4.2. Quadratic Separation at the Boundary -- 4.3. Global Estimates up to the Second Derivatives -- 4.4. Existence of Classical Solutions -- 4.5. The Role of Subsolutions -- 4.6. Pogorelov's Counterexamples to Interior Regularity -- 4.7. Application: The Classical Isoperimetric Inequality -- 4.8. Application: Nonlinear Integration by Parts Inequality -- 4.9. Problems -- 4.10. Notes -- Chapter 5. Sections and Interior First Derivative Estimates -- 5.1. Sections of Convex Functions -- 5.2. Caffarelli's Localization Theorem and Strict Convexity -- 5.3. Interior Hölder Gradient Estimates.