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Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations

VIASM 2016

  • Nam Q. Le
  • Hiroyoshi Mitake
  • Hung V. Tran
  • Hiroyoshi Mitake
  • Hung V. Tran

Part of the Lecture Notes in Mathematics book series (LNM, volume 2183)

Table of contents

  1. Front Matter
    Pages i-vii
  2. The Second Boundary Value Problem of the Prescribed Affine Mean Curvature Equation and Related Linearized Monge-Ampère Equation

    1. Front Matter
      Pages 1-5
    2. Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran
      Pages 7-33
    3. Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran
      Pages 35-72
    4. Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran
      Pages 73-123
  3. Dynamical Properties of Hamilton–Jacobi Equations via the Nonlinear Adjoint Method: Large Time Behavior and Discounted Approximation

    1. Front Matter
      Pages 125-128
    2. Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran
      Pages 129-139
    3. Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran
      Pages 141-176
    4. Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran
      Pages 177-206
    5. Nam Q. Le, Hiroyoshi Mitake, Hung V. Tran
      Pages 207-228
  4. Back Matter
    Pages 229-230

About this book

Introduction

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 new approach 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

Authors and affiliations

  • Nam Q. Le
    • 1
  • Hiroyoshi Mitake
    • 2
  • Hung V. Tran
    • 3
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Institute of EngineeringHiroshima UniversityHigashi-Hiroshima-shiJapan
  3. 3.Department of MathematicsUniversity of Wisconsin MadisonMadisonUSA

Editors and affiliations

  • Hiroyoshi Mitake
    • 1
  • Hung V. Tran
    • 2
  1. 1.Institute of EngineeringHiroshima University Higashi-Hiroshima-shiJapan
  2. 2.Department of MathematicsUniversity of Wisconsin Madison MADISONUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-54208-9
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-54207-2
  • Online ISBN 978-3-319-54208-9
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site