●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

本書繫統清晰地介紹了近年來用弱收斂方法研究非線性偏微分方程的諸多最重要的技術。這項工作是作者於1988年夏天在芝加哥的洛約拉大學（Loyola University）做的十個繫列報告的擴展版本。作者概述了關於不同非線性偏微分方程解的存在性的各項技術，尤其考慮了沒有強解析估計的情況。總體的觀點是，當近似解序列僅能弱收斂時，我們必須利用偏微分方程的非線性結構來驗證計算的極限。作者專注於快速發展的幾個領域，並指出了它們共同的一些基本觀點。本書主題包括：測度論和實分析（與泛函分析相對）的主要作用，以及在各種場合下持續使用低幅、高頻的周期測試函數來提取有用信息。作者通過極簡單的問題來說明各種關鍵技術。本書面向非線性偏微分方程領域的數學研究人員，是理解這一重要研究領域中所使用技術的重要資料。