The monge-ampere equationMonge-Ampere方程

书籍目录

Preface  Notation  1 Generalized Solutions to Monge-Ampere Equations  　1.1 The normal mapping　　1.1.1　Properties of the normal mapping　1.2 Generalized solutions 　1.3 Viscosity solutions 　1.4 Maximum principles　　1.4.1　Aleksandrov's maximum principle 　　1.4.2　Aleksandrov-Bakelman-Pucci's maximum principle 　　1.4.3　Comparison principle 　1.5 The Dirichlet problem 　1.6 The nonhomogeneous Dirichlet problem 　1.7 Return to viscosity solutions 　1.8 Ellipsoids of minimum volume 2 Uniformly Elliptic Equations in Nondivergence Form 　2.1 Critical density estimates 　2.2 Estimate of the distribution function of solutions 　2.3 Harnack's inequality  3 The Cross-sections of Monge-Ampere　3.1 Introduction 　3.2 Preliminary results 　3.3 Properties of the sections　　3.3.1　The Monge-Ampere measures satisfying （3.1.1）　　3.3.2　The engulfing property of the sections 　　3.3.3　The size of normalized sections4 Convex Solutions of det D[superscript 2]u = 1 in R[superscript n] 　4.1 Pogorelov's Lemma　4.2 Interior Holder estimates of D[superscript 2]u u 5 Regularity Theory for the Monge-Ampere Equation 　5.1 Extremal points 　5.2 A result on extremal points of zeroes of solutions to Monge-Ampere 　5.3 A strict convexity result 　5.4 C[superscript 1,[alpha]] regularity 　5.5 Examples 6 W[superscript 2,p] Estimates for the Monge-Ampere Equation 　6.1 Approximation Theorem 　6.2 Tangent paraboloids 　6.3 Density estimates and power decay 　6.4 L[superscript p] estimates of second derivatives 　6.5 Proof of the Covering Theorem 6.3.3 　6.6 Regularity of the convex envelopeBibliography Index

作者简介

The Monge-Ampere equation has attracted considerable interest in recent years because of its important role in several areas of applied mathematics. Monge-Ampere type equations have applications in the areas of differential geometry, the calculus of variations, and several optimization problems, such as the Monge-Kantorovitch mass transfer problem. This book stresses the geometric aspects of this beautiful theory, using techniques from harmonic analysis - covering lemmas and set decompositions.

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