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

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 MongeAmpère and linearized MongeAmpère 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 MongeAmpère 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 HamiltonJacobi 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 HamiltonJacobi equations.

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

Inhalt
Preface by Nguyen Huu Du (Managing director of VIASM).-M iroyoshi Mitake ***and Hung V. Tran: Dynamical properties of Hamilton-Jacobi equations via the nonlinear adjoint method: Large time behavior and Discounted approximation.- Nam Q. Le*: The second boundary value problem of the prescribed affine mean curvature equation and related linearized Monge-Ampère equation.