Dynamical and geometric aspects of Hamilton-Jacobi and linearized Monge-Ampére equations. VIASM 2016. Based on lectures at the Rutgers University, NJ, USA, 2013. (English) Zbl 1373.35006

Mitake, Hiroyoshi (ed.); Tran, Hung V. (ed.)

Lecture Notes in Mathematics 2183. Cham: Springer (ISBN 978-3-319-54207-2/pbk; 978-3-319-54208-9/ebook). vii, 228 p. (2017).

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Publisher’s description: 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 Kähler 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.
Part I and II of this volume will be reviewed individually.

MSC:

35-06	Proceedings, conferences, collections, etc. pertaining to partial differential equations
35B10	Periodic solutions to PDEs
35B40	Asymptotic behavior of solutions to PDEs
35B50	Maximum principles in context of PDEs
35B51	Comparison principles in context of PDEs
35B65	Smoothness and regularity of solutions to PDEs
35J40	Boundary value problems for higher-order elliptic equations