Direct methods in the calculus of variations

Contents
Introduction
Chapter 1 Semi-Classical Theory
1.1 The Maximum Principle
1.2 The Bounded Slope Condition
1.3 Barriers
1.4 The Area Functional
1.5 Non-Existence of Minimal Surfaces
1.6 Notes and Comments
Chapter 2 Measurable Functions
2.1 Lp Spaces
2.2 Test Functions and Mollifiers
2.3 Morrey's and Campanato's Spaces
2.4 The Lemmas of John and Nirenberg
2.5 Interpolation
2.6 The Hausdorff Measure
2.7 Notes and Comments
Chapter 3 Sobolev Spaces
3.1 Partitions of Unity
3.2 Weak Derivatives
3.3 The Sobolev Spaces Wkp
3.4 Imbedding Theorems
3.5 Compactness
3.6 Inequalities
3.7 Traces
3.8 The Values of W1p Functions
3.9 Notes and Comments
Chapter 4 Convexity and Semicontinuity
4.1 Preliminaries
4.2 Convex Functional
4.3 Semicontinuity
4.4 An Existence Theorem
4.5 Notes and Comments
Chapter 5 Quasi-Convex Functionals
5.1 Necessary Conditions
5.2 First Semicontinuity Results
5.3 The Quasi-Convex Envelope
5.4 The Ekeland Variational Principle
5.5 Semicontinuity
5.6 Coerciveness and Existence
5.7 Notes and Comments
Chapter 6 Quasi-Minima
6.1 Preliminaries
6.2 Quasi-Minima and Differential Quations
6.3 Cubical Quasi-Minima
6.4 Lp Estimates for the Gradient
6.5 Boundary Estimates
6.6 Notes and Comments
Chapter 7 Holder Continuity
7.1 Caccioppoli's Inequality
7.2 De Giorgi Classes
7.3 Quasi-Minima
7.4 Boundary Regularity
7.5 The Harnack Inequality
7.6 The Homogeneous Case
7.7 w-Minima
7.8 Boundary Regularity
7.9 Notes and Comments
Chapter 8 First Derivatives
8.1 The Difference Quotients
8.2 Second Derivatives
8.3 Gradient Estimates
8.4 Boundary Estimates
8.5 w-Minima
8.6 Holder Continuity of the Derivatives (p=2)
8.7 Other Gradient Estimates
8.8 Holder Continuity of the Derivatives (p#2)
8.9 Elliptic Equations
8.10 Notes and Comments
Chapter 9 Partial Regularity
9.1 Preliminaries
9.2 Quadratic Functional
9.3 The Second Caccioppoli Inequality
9.4 The Case F = F(z) (p = 2)
9.5 Partial Regularity
9.6 Notes and Comments
Chapter 10 Higher Derivatives
10.1 Hilbert Regularity
10.2 Constant Coefficients
10.3 Continuous Coefficients
10.4 Lp Estimates
10.5 Minima of Functionals
10.6 Notes and Comments
References
Index