Analysis of Monge–Ampère Equations

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
		5.4. Counterexamples to Hölder Gradient Estimates
		5.5. Geometry of Sections
		5.6. Vitali Covering and Crawling of Ink-spots Lemmas
		5.7. Centered Sections
		5.8. Problems
		5.9. Notes
	Chapter 6. Interior Second Derivative Estimates
		6.1. Pogorelov’s Second Derivative Estimates
		6.2. Second-Order Differentiability of the Convex Envelope
		6.3. Density Estimate
		6.4. Interior Second Derivative Sobolev Estimates
		6.5. Counterexamples of Wang and Mooney
		6.6. Interior Second Derivative Hölder Estimates
		6.7. Further Remarks on Pogorelov-type Estimates
		6.8. Problems
		6.9. Notes
	Chapter 7. Viscosity Solutions and Liouville-type Theorems
		7.1. Viscosity Solutions of the Monge–Ampère Equation
		7.2. Viscosity Solutions with Boundary Discontinuities
		7.3. Liouville-type Theorems
		7.4. Application: Connections between Four Important PDEs in Two Dimensions
		7.5. Legendre–Lewy Rotation
		7.6. Problems
		7.7. Notes
	Chapter 8. Boundary Localization
		8.1. Preliminary Localization Property of Boundary Sections
		8.2. Savin’s Boundary Localization Theorem
		8.3. Normalized Altitude of Boundary Sections
		8.4. Proof of the Boundary Localization Theorem
		8.5. Pointwise Hölder Gradient Estimates at the Boundary
		8.6. Boundary Localization for Degenerate and Singular Equations
		8.7. Problems
		8.8. Notes
	Chapter 9. Geometry of Boundary Sections
		9.1. Maximal Interior Sections and Rescalings
		9.2. Global Hölder Gradient Estimates
		9.3. Dichotomy, Volume Growth, and Engulfing Properties
		9.4. Global Inclusion, Exclusion, and Chain Properties
		9.5. Besicovitch’s Covering Lemma, Maximal Function, and Quasi-Distance
		9.6. Problems
		9.7. Notes
	Chapter 10. Boundary Second Derivative Estimates
		10.1. Global Second Derivative Sobolev Estimates
		10.2. Wang’s Counterexamples
		10.3. Boundary Pogorelov Second Derivative Estimates
		10.4. Pointwise Second Derivative Hölder Estimates at the Boundary
		10.5. Global Second Derivative Hölder Estimates
		10.6. Problems
		10.7. Notes
	Chapter 11. Monge–Ampère Eigenvalue and Variational Method
		11.1. The Monge–Ampère Energy
		11.2. Parabolic Monge–Ampère Flow
		11.3. Degenerate Monge–Ampère Equations
		11.4. Monge–Ampère Eigenvalue and Eigenfunctions
		11.5. Global Regularity for Degenerate Equations
		11.6. Problems
		11.7. Notes
Part 2. The Linearized Monge–Ampère Equation
	Chapter 12. Interior Harnack Inequality
		12.1. Caffarelli–Gutiérrez Harnack Inequality
		12.2. Proof of the Interior Harnack Inequality
		12.3. Interior Hölder Estimates
		12.4. Application: The Affine Bernstein Problem
		12.5. Problems
		12.6. Notes
	Chapter 13. Boundary Estimates
		13.1. Global Hölder Estimates
		13.2. Application: Solvability of Affine Mean Curvature and Abreu’s Equations
		13.3. Boundary Gradient Estimates
		13.4. Boundary Hölder Gradient Estimates
		13.5. Boundary Harnack Inequality
		13.6. Application: Minimizers of Linear Functionals with Prescribed Determinant
		13.7. Problems
		13.8. Notes
	Chapter 14. Green’s Function
		14.1. Bounds and Higher Integrability
		14.2. Local Integrability Estimates for the Gradient
		14.3. Monge–Ampère Sobolev Inequality
		14.4. Global Higher Integrability Estimates
		14.5. Global Integrability Estimates for the Gradient
		14.6. Application: Hölder Estimates for Equations with Critical Inhomogeneity
		14.7. Problems
		14.8. Notes
	Chapter 15. Divergence Form Equations
		15.1. Interior Uniform Estimates via Moser Iteration
		15.2. Interior Hölder estimates
		15.3. Global Hölder Estimates
		15.4. Application: The Dual Semigeostrophic Equations
		15.5. Application: Singular Abreu Equations
		15.6. Problems
		15.7. Notes
	Bibliography
	Index
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