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The Monge—Ampère Equation

  • Cristian E. Gutiérrez

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

Table of contents

  1. Front Matter
    Pages i-xi
  2. Cristian E. Gutiérrez
    Pages 1-30
  3. Cristian E. Gutiérrez
    Pages 31-43
  4. Cristian E. Gutiérrez
    Pages 45-62
  5. Cristian E. Gutiérrez
    Pages 63-74
  6. Cristian E. Gutiérrez
    Pages 75-93
  7. Cristian E. Gutiérrez
    Pages 95-122
  8. Back Matter
    Pages 123-132

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  • Cristian E. Gutiérrez
    • 1
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA

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