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.
