Now in its second edition this monograph explores the Monge-Ampère equation and the latest
advances in its study and applications. It provides an essentially self-contained systematic
exposition of the theory of weak solutions including regularity results by L. A. Caffarelli.
The geometric aspects of this theory are stressed using techniques from harmonic analysis such
as covering lemmas and set decompositions. An effort is made to present complete proofs of all
theorems and examples and exercises are offered to further illustrate important concepts. Some
of the topics considered include generalized solutions non-divergence equations cross
sections and convex solutions. New to this edition is a chapter on the linearized Monge-Ampère
equation and a chapter on interior Hölder estimates for second derivatives. Bibliographic notes
updated and expanded from the first edition are included at the end of every chapter for
further reading on Monge-Ampère-type equations and their diverse applications in the areas of
differential geometry the calculus of variations optimization problems optimal mass
transport and geometric optics. Both researchers and graduate students working on nonlinear
differential equations and their applications will find this to be a useful and concise
resource.
