Authors:
Editors:
- Contains a gentle introduction to Monge-Ampère equations
- Offers a starting point to learn the theory of viscosity solutions (see appendix of part 2)
- Provides up-to-date research directions in the fields of Hamilton-Jacobi and linearized Monge-Ampere equations
- Includes supplementary material: sn.pub/extras
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2183)
Buy it now
Buying options
Tax calculation will be finalised at checkout
Other ways to access
This is a preview of subscription content, log in via an institution to check for access.
Table of contents (7 chapters)
-
Front Matter
-
The Second Boundary Value Problem of the Prescribed Affine Mean Curvature Equation and Related Linearized Monge-Ampère Equation
-
Front Matter
-
-
Dynamical Properties of Hamilton–Jacobi Equations via the Nonlinear Adjoint Method: Large Time Behavior and Discounted Approximation
-
Front Matter
-
-
Back Matter
About this book
Consisting of two parts, the first part of this volume is an essentially self-contained exposition of the geometric aspects of local and global regularity theory for the Monge–Ampère and linearized Monge–Ampère equations. As an application, we solve the second boundary value problem of the prescribed affine mean curvature equation, which can be viewed as a coupling of the latter two equations. Of interest in its own right, the linearized Monge–Ampère equation also has deep connections and applications in analysis, fluid mechanics and geometry, including the semi-geostrophic equations in atmospheric flows, the affine maximal surface equation in affine geometry and the problem of finding Kahler metrics of constant scalar curvature in complex geometry.
Among other topics, the second part provides a thorough exposition of the large time behavior and discounted approximation of Hamilton–Jacobi equations, which have received much attention in the last two decades, and a newapproach to the subject, the nonlinear adjoint method, is introduced. The appendix offers a short introduction to the theory of viscosity solutions of first-order Hamilton–Jacobi equations.
Keywords
- 35B10,35B27,35B40, 35B45,35B50,35B51,35B65,35D40,35J40,
- Hamilton-Jacobi equations
- Monge-Ampere equations
- linearized Monge-Ampere equations
- large time behavior
- selection problem
- affine mean curvature equation
- second boundary value problem
- introduction to the theory of viscosity solutions
- Caffarelli-Guti´errez Harnack inequality
- affine Bernstein problem
- partial differential equations
Authors, Editors and Affiliations
-
Institute of Engineering, Hiroshima University , Higashi-Hiroshima-shi, Japan
Hiroyoshi Mitake
-
Department of Mathematics, University of Wisconsin Madison , MADISON, USA
Hung V. Tran
-
Department of Mathematics, Indiana University, Bloomington, USA
Nam Q. Le
-
Institute of Engineering, Hiroshima University, Higashi-Hiroshima-shi, Japan
Hiroyoshi Mitake
-
Department of Mathematics, University of Wisconsin Madison, Madison, USA
Hung V. Tran
Bibliographic Information
Book Title: Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations
Book Subtitle: VIASM 2016
Authors: Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran
Editors: Hiroyoshi Mitake, Hung V. Tran
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-54208-9
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2017
Softcover ISBN: 978-3-319-54207-2Published: 16 June 2017
eBook ISBN: 978-3-319-54208-9Published: 14 June 2017
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: VII, 228
Number of Illustrations: 15 b/w illustrations, 1 illustrations in colour
Topics: Partial Differential Equations, Calculus of Variations and Optimal Control; Optimization, Differential Geometry