非线性偏微分方程的弱收敛方法

本书系统清晰地介绍了近年来用弱收敛方法研究非线性偏微分方程的诸多最重要的技术。这项工作是作者于1988年夏天在芝加哥的洛约拉大学（Loyola University）做的十个系列报告的扩展版本。作者概述了关于不同非线性偏微分方程解的存在性的各项技术，尤其考虑了没有强解析估计的情况。总体的观点是，当近似解序列仅能弱收敛时，我们必须利用偏微分方程的非线性结构来验证计算的极限。作者专注于快速发展的几个领域，并指出了它们共同的一些基本观点。本书主题包括：测度论和实分析（与泛函分析相对）的主要作用，以及在各种场合下持续使用低幅、高频的周期测试函数来提取有用信息。作者通过极简单的问题来说明各种关键技术。本书面向非线性偏微分方程领域的数学研究人员，是理解这一重要研究领域中所使用技术的重要资料。

Preface
Introduction
1.Weak Convergence
A.Review of Basic Theory
B.Convergence of Averages
C.Compactness in Sobolev Spaces
1.Embeddings
2.Compactness Theorems
3.A Refinement of Rellich's Theorem
D.Measures of Concentration
1.Generalities
2.Defect Measures
3.A Refinement of Fatou's Lemma
4.Concentration and Sobolev Inequalities
E.Measures of Oscillation
1.Generalities
2.Slicing Measures
3.Young Measures
2.Convexity
A.The Calculus of Variations
B.Weak Lower Semicontinuity
C.Convergence of Energies and Strong Convergence
3.Quasiconvexity
A.Definitions
1.Rank-One Convexity
2.Quasiconvexity
B.Weak Lower Semicontinuity
C.Convergence of Energies and Strong Convergence
D.Partial Regularity of Minimizers
E.Examples
1.Weak Continuity of Determinants
2.Polyconvexity
4.Concentrated Compactness
A.Variational Problems
1.Minimizers for Critical Sobolev Nonlinearities
2.Strong Convergence of Minimizing Sequences
B.Concentration-Cancellation
1.Critical Gradient Growth
2.Vorticity Bounds and Euler's Equations
5.Compensated Compactness
A.Direct Methods
1.Harmonic Maps into Spheres
2.Homogenization of Divergence Structure PDE's
3.Monotonicity,Minty-Browder Method in L2
B.Div-Curl Lemma
C.Elliptic Systems
D.Conservation Laws
1.Single Equations
2.Systems of Two Equations
E.Generalization of Div-Curl Lemma
6.Maximum Principle Methods
A.The Maximum Principle for Fully Nonlinear PDE
1.Minty-Browder Method in L∞
2.Viscosity Solutions
B.Homogenization of Nondivergence Structure PDE's
C.Singular Perturbations
Appendix
Notes
References