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The Monge—Ampère Equation

  • Book
  • © 2001

Overview

Authors:
  1. Cristian E. Gutiérrez

    1. Department of Mathematics, Temple University, Philadelphia, USA

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

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xi
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  2. Generalized Solutions to Monge-Ampère Equations

    • Cristian E. Gutiérrez
    Pages 1-30

  3. Uniformly Elliptic Equations in Nondivergence Form

    • Cristian E. Gutiérrez
    Pages 31-43

  4. The Cross-sections of Monge—Ampère

    • Cristian E. Gutiérrez
    Pages 45-62

  5. Convex Solutions of detD2u = 1 in ℝn

    • Cristian E. Gutiérrez
    Pages 63-74

  6. Regularity Theory for the Monge-Ampère Equation

    • Cristian E. Gutiérrez
    Pages 75-93

  7. W2,p Estimates for the Monge—Amperè Equation

    • Cristian E. Gutiérrez
    Pages 95-122

  8. Back Matter

    Pages 123-132
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Temple University, Philadelphia, USA

    Cristian E. Gutiérrez

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