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Full Description
This book presents a systematic analysis of the Monge-Ampere equation, the linearized Monge-Ampere 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-Ampere 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-Ampere 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 Holder 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.
Contents
Introduction
Geometric and analytic preliminaries
The Monge-Ampere equation: Aleksandrov solutions and maximum principles
Classical solutions
Sections and interior first derivative estimates
Interior second derivative estimates
Viscosity solutions and Liouville-type theorems
Boundary localization
Geometry of boundary sections
Boundary second derivative estimates
Monge-Ampere eigenvalue and variational method
The linearized Monge-Ampere equation: Interior Harnack inequality
Boundary estimates
Green's function
Divergence form equations
Bibliography
Index.
