The Monge-Ampère Equation and Its Applications

This book arose through a series of lectures at ETH Zürich on the Monge-Ampère equation and its applications (as the title aptly suggests). After a thorough reading of this book, one is quite near to the forefront on recent results concerning solvability of the Monge-Ampère equation. One of my favorite parts of the book was actually the appendix, where precise statements (and proofs!) of required results are given. This allows for a nice flow of the material in the book, and yet the precise statements given in the appendix are quite useful in and of themselves. The author also does an exceptional job by including many figures throughout the text. This was extremely useful in understanding some of the more technical arguments.

Overall I found the book to be extremely well written, with very precise statements and clear proofs. One aspect I really liked was the fact that most constants arising in various estimates are written out explicitly. Many authors do not bother with this, but somehow I believe it adds quite a lot to the exposition. The longer proofs are split into various steps, which helps to digest the long arguments.

The first chapter analyzes existence and uniqueness of Alexandrov solutions (a type of generalized solution) for the Dirichlet problem. Chapter 3 then addresses smoothness of such solutions when the underlying domain and boundary data are smooth. Chapter 3 also addresses a counterexample of Pogorelov (Section 3.3), with explicit, understandable calculations. Chapter 4 is particularly nice as a number of concrete applications are given, ranging from optimal transport applications to the semigeostrophic equations. Finally, Chapter 5 ends nicely with extensions of previous results.

This book would serve as a great reference for a graduate topics course on the Monge-Ampère equation. However, it would only serve as a reference, since there are no exercises provided. This book would additionally serve quite well as a reference for research level mathematics on nonlinear partial differential equations.

Eric Stachura is currently a Visiting Assistant Professor of Mathematics at Haverford College.