# The Monge—Ampère Equation

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 44)

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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.

PDEs application differential geometry harmonic analysis linear optimization maximum principle nonlinear equations optimization

- DOI https://doi.org/10.1007/978-1-4612-0195-3
- Copyright Information Birkhäuser Boston 2001
- Publisher Name Birkhäuser, Boston, MA
- eBook Packages Springer Book Archive
- Print ISBN 978-1-4612-6656-3
- Online ISBN 978-1-4612-0195-3
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