This updated and revised monograph continues to follow the latest advances in the study of the Monge-Ampère equation and its applications. These advances are reflected in an essentially self-contained systematic exposition of the theory of weak solutions, including recent regularity results by L.A. Caffarelli. This volume can be used for a graduate level topics course in differential equations, and features bibliographic notes at the end of each chapter for further exploration.
Additions to the second edition include:
- A new proof of the theorem that viscosity solutions are Aleksandrov solutions without using deep regularity results
- A new chapter on the Harnack inequality for the linearized Monge-Ampère equation.
- A new chapter on interior Hölder estimates for second derivatives
- Note sections expanded to include new developments since 2001
