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