Introduction .................................................... 1
Chapter 1 Semi-Classical Theory ............................... 13
1.1 The Maximum Principle ..................................... 14
1.2 The Bounded Slope Condition ............................... 18
1.3 Barriers .................................................. 22
1.4 The Area Functional ....................................... 30
1.5 Non-Existence of Minimal Surfaces ......................... 32
1.6 Notes and Comments ........................................ 36
Chapter 2 Measurable Functions ................................ 39
2.1 Lp Spaces ................................................. 39
2.2 Test Functions and Mollifiers ............................. 44
2.3 Morrey's and Campanato's Spaces ........................... 46
2.4 The Lemmas of John and Nirenberg .......................... 54
2.5 Interpolation ............................................. 61
2.6 The Hausdorff Measure ..................................... 68
2.7 Notes and Comments ........................................ 71
Chapter 3 Sobolev Spaces ...................................... 75
3.1 Partitions of Unity ....................................... 75
3.2 Weak Derivatives .......................................... 79
3.3 The Sobolev Spaces Wk,p ................................... 83
3.4 Imbedding Theorems ........................................ 90
3.5 Compactness ............................................... 98
3.6 Inequalities ............................................. 101
3.7 Traces ................................................... 106
3.8 The Values of W1,p Functions ............................. 110
3.9 Notes and Comments ....................................... 114
Chapter 4 Convexity and Semicontinuity ....................... 119
4.1 Preliminaries ............................................ 119
4.2 Convex Functional ........................................ 121
4.3 Semicontinuity ........................................... 123
4.4 An Existence Theorem ..................................... 131
4.5 Notes and Comments ....................................... 134
Chapter 5 Quasi-Convex Functionals ........................... 139
5.1 Necessary Conditions ..................................... 139
5.2 First Semicontinuity Results ............................. 150
5.3 The Quasi-Convex Envelope ................................ 155
5.4 The Ekeland Variational Principle ........................ 160
5.5 Semicontinuity 4 ......................................... 162
5.6 Coerciveness and Existence ............................... 168
5.7 Notes and Comments ....................................... 170
Chapter 6 Quasi-Minima ....................................... 173
6.1 Preliminaries ............................................ 173
6.2 Quasi-Minima and Differential Quations ................... 175
6.3 Cubical Quasi-Minima ..................................... 187
6.4 Lp Estimates for the Gradient ............................ 197
6.5 Boundary Estimates ....................................... 206
6.6 Notes and Comments ....................................... 209
Chapter 7 Holder Continuity .................................. 213
7.1 Caccioppoli's Inequality ................................. 213
7.2 De Giorgi Classes ........................................ 218
7.3 Quasi-Minima ............................................. 225
7.4 Boundary Regularity ...................................... 232
7.5 The Harnack Inequality ................................... 235
7.6 The Homogeneous Case ..................................... 242
7.7 ω-Minima ................................................. 245
7.8 Boundary Regularity ...................................... 255
7.9 Notes and Comments ....................................... 257
Chapter 8 First Derivatives .................................. 261
8.1 The Difference Quotients ................................. 263
8.2 Second Derivatives ....................................... 266
8.3 Gradient Estimates ....................................... 271
8.4 Boundary Estimates ....................................... 274
8.5 ω-Minima ................................................. 278
8.6 Holder Continuity of the Derivatives (p = 2) ............. 285
8.7 Other Gradient Estimates ................................. 288
8.8 Holder Continuity of the Derivatives (p ≠ 2) ............. 298
8.9 Elliptic Equations ....................................... 301
8.10 Notes and Comments ....................................... 304
Chapter 9 Partial Regularity ................................. 307
9.1 Preliminaries ............................................ 307
9.2 Quadratic Functional ..................................... 309
9.3 The Second Caccioppoli Inequality ........................ 319
9.4 The Case F = F(z) (p = 2) ................................ 329
9.5 Partial Regularity ....................................... 333
9.6 Notes and Comments ....................................... 342
Chapter 10 Higher Derivatives ................................. 347
10.1 Hilbert Regularity ....................................... 348
10.2 Constant Coefficients .................................... 355
10.3 Continuous Coefficients .................................. 362
10.4 LP Estimates ............................................. 368
10.5 Minima of Functional ..................................... 374
10.6 Notes and Comments ....................................... 375
References .................................................... 377
Index ......................................................... 399
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