The monge-ampere equationMonge-Ampere方程

內(nèi)容概要

The classical Monge-Ampère equation has been the center of considerable interest in recent years because of its important role in several areas of applied mathematics。In reflecting these developments，this works stresses the geometric aspects of this beautiful theory，using some techniques from harmonic analysis covering lemmas and set decompositions。Moreover，Monge-Ampère 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。The book is an essentially self-contained exposition of the theory of weak solutions，including the regularity results of L.A。Caffarelli。The presentation unfolds systematically from introductory chapters，and an effort is made to present complete proofs of all theorems。Included are examples，illustrations，bibliographical references at the end of each chapter，and a comprehensive index。Topics covered include: Generalized Solutions * Non-divergence Equations * The Cross-Sections of Monge-Ampère Convex Solutions of D^2u = 1 in R^n * Regularity Theory * W^2，p Estimates The Monge-Ampère Equation is a concise and useful book for graduate students and researchers in the field of nonlinear equations。

書籍目錄

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

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