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Analysis of Monge–Ampère Equations
 
Nam Q. Le Indiana University, Bloomington, IN
Hardcover ISBN:  978-1-4704-7420-1
Product Code:  GSM/240
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
Softcover ISBN:  978-1-4704-7625-0
Product Code:  GSM/240.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-7624-3
Product Code:  GSM/240.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7625-0
eBook: ISBN:  978-1-4704-7624-3
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
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Analysis of Monge–Ampère Equations
Nam Q. Le Indiana University, Bloomington, IN
Hardcover ISBN:  978-1-4704-7420-1
Product Code:  GSM/240
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
Softcover ISBN:  978-1-4704-7625-0
Product Code:  GSM/240.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-7624-3
Product Code:  GSM/240.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7625-0
eBook ISBN:  978-1-4704-7624-3
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 Details
     
     
    Graduate Studies in Mathematics
    Volume: 2402024; 576 pp
    MSC: 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 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.

    Readership

    Graduate students and researchers interested in non-linear 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 Liouville-type 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 2402024; 576 pp
MSC: 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 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.

Readership

Graduate students and researchers interested in non-linear 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 Liouville-type 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
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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