本书的主要目的是全面阐述作者关于发散形式的二阶椭圆拟线性方程弱解的边界正则性的相关工作成果。这些方程的结构容许系数在特定的Lp空间中，因此从经典结果可知，弱解在内部是局部H?lder连续的。这里表明了，弱解在边界处是连续的当且仅当Wiener型条件得到满足。 在调和函数的情形下，这个条件约化为著名的Wiener准则。这个分析的过程还包括对Sobolev空间的“精细”分析以及相关非线性位势论的研究。术语“精细”是指由Wiener条件诱导的Rn的拓扑结构。
本书还完整讲述了变分不等式的解的正则性，包括双障碍问题，其中障碍可以是不连续的。 解的正则性涉及Wiener型条件并以精细拓扑结构的形式给出。 本书还讨论了具有可微结构和C1,α障碍的微分算子的情形。书中的一章专门讨论了存在理论，从而为读者提供了从弱解的正则性到弱解的存在性的完整处理。
本书适合于对椭圆微分方程弱解的正则性理论、Sobolev空间和位势论感兴趣的研究生阅读，也可供相关研究人员参考。

Preface
Basic Notation
Chapter 1. Preliminaries
1.1 Basic results
1.1.1 Covering theorems
1.1.2 Densities of measures
1.1.3 The maximal function and its applications
1.2 Potential estimates
1.3 Sobolev spaces
1.3.1 Inequalities
1.3.2 Imbeddings
1.3.3 Pointwise differentiability of Sobolev functions
1.3.4 Spaces y1,p
1.3.5 Adams' inequality
1.3.6 Bessel and Riesz potentials
1.4 Historical notes
Chapter 2. Potential Theory
2.1 Capacity
2.1.1 Comparison of capacities; capacities of balls
2.1.2 Polar sets
2.1.3 Quasicontinuity
2.1.4 Multipliers
2.1.5 Capacity and energy minimizers
2.1.6 Thinness
2.1.7 Capacity and Hausdorff measure
2.1.8 Lebesgue points for Sobolev functions
2.2 Laplace equation
2.2.1 Green potentials
2.2.2 Classical thinness
2.2.3 Dirichlet problem and the Wiener criterion
2.3 Regularity of minimizers
2.3.1 Abstract minimization
2.3.2 Minimizers and weak solutions
2.3.3 Higher regularity
2.3.4 The De Giorgi method
2.3.5 Moser's iteration technique
2.3.6 Removable singularities
2.3.7 Estimates of supersolutions
2.3.8 Estimates of energy minimizers
2.3.9 Dirichlet problem
2.3.10 Application of thinness: the Wiener criterion
2.4 Fine topology
2.5 Fine Sobolev spaces
2.6 Historical notes
Chapter 3. Quasilinear Equations
3.1 Basic properties of weak solutions
3.1.1 Upper bounds for weak solutions
3.1.2 Weak Harnack inequality
3.1.3 Removable sets for weak solutions
3.2 Higher regularity of equations with differentiable structure
3.3 Historical notes
Chapter 4. Fine Regularity Theory
4.1 Basic energy estimates
4.2 Sufficiency of the Wiener condition for boundary regularity
4.2.1 The special case of harmonic functions
4.3 Necessity of the Wiener condition for boundary regularity
4.3.1 Main estimate
4.3.2 Necessity of the Wiener condition
4.4 Equations with measure data
4.5 Historical notes
Chapter 5. Variational Inequalities - Regularity
5.1 Differential operators with measurable coefficients
5.1.1 Continuity in the presence of irregular obstacles
5.1.2 The modulus of continuity
5.2 Differential operators with differentiable structure
5.3 Historical notes
Chapter 6. Existence Theory
6.1 Existence of solutions to variational inequalities
6.1.1 Pseudomonotone operators
6.1.2 Variational problems - existence of bounded solutions
6.1.3 Variational problems leading to unbounded solutions
6.2 The Dirichlet problem for equations with differentiable structure
6.3 Historical notes
References
Index
Notation Index