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Softcover ISBN:  9781470476250 
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Hardcover ISBN:  9781470474201 
Product Code:  GSM/240 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
Softcover ISBN:  9781470476250 
Product Code:  GSM/240.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470476243 
Product Code:  GSM/240.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470476250 
eBook ISBN:  9781470476243 
Product Code:  GSM/240.S.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsGraduate Studies in MathematicsVolume: 240; 2024; 576 ppMSC: Primary 35; Secondary 52; 49
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 selfcontained 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.
ReadershipGraduate students and researchers interested in nonlinear PDE.

Table of Contents

Chapters

Introduction

Geometric and analytic preliminaries

The Monge–Ampère equation

Aleksandrov solutions and maximum principles

Classical solutions

Sections and interior first derivative estimates

Interior second derivative estimates

Viscosity solutions and Liouvilletype theorems

Boundary localization

Geometry of boundary sections

Boundary second derivative estimates

Monge–Ampère eigenvalue and variational method

The linearized Monge–Ampère equation

Interior Harnack inequality

Boundary estimates

Green’s function

Divergence form equations


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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 selfcontained 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.
Graduate students and researchers interested in nonlinear PDE.

Chapters

Introduction

Geometric and analytic preliminaries

The Monge–Ampère equation

Aleksandrov solutions and maximum principles

Classical solutions

Sections and interior first derivative estimates

Interior second derivative estimates

Viscosity solutions and Liouvilletype theorems

Boundary localization

Geometry of boundary sections

Boundary second derivative estimates

Monge–Ampère eigenvalue and variational method

The linearized Monge–Ampère equation

Interior Harnack inequality

Boundary estimates

Green’s function

Divergence form equations