The Monge—Ampère Equation

Cristian E. Gutiérrez (auth.)0817641777, 9780817641771, 3764341777, 9783764341770

In recent years, the study of the Monge-Ampere equation has received consider­ able attention and there have been many important advances. As a consequence there is nowadays much interest in this equation and its applications. This volume tries to reflect these advances in an essentially self-contained systematic exposi­ tion of the theory of weak: solutions, including recent regularity results by L. A. Caffarelli. The theory has a geometric flavor and uses some techniques from har­ monic analysis such us covering lemmas and set decompositions. An overview of the contents of the book is as follows. We shall be concerned with the Monge-Ampere equation, which for a smooth function u, is given by (0.0.1) There is a notion of generalized or weak solution to (0.0.1): for u convex in a domain n, one can define a measure Mu in n such that if u is smooth, then Mu 2 has density det D u. Therefore u is a generalized solution of (0.0.1) if M u = f.

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Table of contents :
Front Matter….Pages i-xi
Generalized Solutions to Monge-Ampère Equations….Pages 1-30
Uniformly Elliptic Equations in Nondivergence Form….Pages 31-43
The Cross-sections of Monge—Ampère….Pages 45-62
Convex Solutions of det D 2 u = 1 in ℝ n ….Pages 63-74
Regularity Theory for the Monge-Ampère Equation….Pages 75-93
W 2,p Estimates for the Monge—Amperè Equation….Pages 95-122
Back Matter….Pages 123-132