Preface ......................................................... v
1 Newtonian spaces ............................................. 1
1.1 The metric space X and some notation .................... 1
1.2 Preliminaries ........................................... 3
1.3 Upper gradients and the Newtonian space N1,p ............ 8
1.4 The Sobolev capacity Cp ................................ 12
1.5 p -weak upper gradients and modulus of curve families .. 15
1.6 Banach space and ACCp .................................. 24
1.7 Examples ............................................... 31
1.8 Notes .................................................. 34
2 Minimal p-weak upper gradients .............................. 37
2.1 Fuglede's lemma ........................................ 37
2.2 Minimal p-weak upper gradients ......................... 40
2.3 Calculus for p-weak upper gradients .................... 45
2.4 The glueing lemma ...................................... 48
2.5 N1,p(Ω) ................................................ 50
2.6 Nloc1,p and Dlocp ....................................... 52
2.7 N01,p .................................................. 55
2.8 Glocp .................................................. 58
2.9 Dependence on p in gu .................................. 59
2.10 Representation formulas for gu ......................... 61
2.11 Notes .................................................. 64
3 Doubling measures ........................................... 65
3.1 Doubling ............................................... 65
3.2 The maximal function ................................... 68
3.3 BMO and John-Nirenberg's lemma ......................... 70
3.4 Consequences of John-Nirenberg's lemma ................. 75
3.5 Gehring's lemma ........................................ 77
3.6 Notes .................................................. 82
4 Poincare inequalities ....................................... 84
4.1 Poincare inequalities .................................. 84
4.2 Characterizations of Poincare inequalities ............. 88
4.3 BiLipschitz invariance ................................. 89
4.4 (q, p)-Pomcai6 inequalities ............................ 91
4.5 Quasiconvexity and connectivity ........................ 99
4.6 Poincare inequalities in quasiconvex spaces ........... 103
4.7 Inner metric .......................................... 106
4.8 The relation between L and λ .......................... 110
4.9 Measurability ......................................... 113
4.10 Notes ................................................. 113
5 Properties of Newtonian functions .......................... 116
5.1 Density of Lipschitz functions ........................ 116
5.2 Quasicontinuity of Newtonian functions ................ 123
5.3 Continuity of Newtonian functions ..................... 134
5.4 Density of Lipschitz functions in N01,p ............... 138
5.5 Sobolev embeddings and inequalities ................... 141
5.6 Lebesgue points for N1,p-functions .................... 147
5.7 Notes ................................................. 150
6 Capacities ................................................. 154
6.1 Mazur's lemma and its consequences .................... 154
6.2 Properties of Cp in complete doubling p-Poincare
spaces ................................................ 157
6.3 The variational capacity capp ......................... 161
6.4 Notes ................................................. 168
7 Superminimizers ............................................ 170
7.1 Introduction to potential theory ...................... 170
7.2 The obstacle problem .................................. 172
7.3 Definition of (super)minimizers ....................... 178
7.4 Convergence results for superminimizers ............... 183
7.5 Notes ................................................. 189
8 Interior regularity ........................................ 191
8.1 Weak Harnack inequalities for subminimizers ........... 191
8.2 Weak Harnack inequalities for superminimizers ......... 197
8.3 Holder continuity for λ-harmonic functions ............ 201
8.4 The need for λ in Harnack's inequality ................ 205
8.5 Lsc-regularized superminimizers ....................... 206
8.6 Lsc-regularized solutions of obstacle problems ........ 209
8.7 p-harmonic extensions ................................. 212
8.8 A sharp weak Harnack inequality for superminimizers ... 213
8.9 Notes ................................................. 215
9 Superharmonic functions .................................... 218
9.1 Definition of superharmonic functions ................. 218
9.2 Weak Harnack inequalities for superharmonic
functions ............................................. 220
9.3 Lsc-regularity and the minimum principle .............. 222
9.4 Characterizations ..................................... 226
9.5 Convergence results for superharmonic functions ....... 229
9.6 Harnack's convergence theorem for p-harmonic
functions ............................................. 233
9.7 Comparison of sub- and superharmonic functions ........ 234
9.8 New superharmonic functions from old .................. 235
9.9 Integrability of superharmonic functions .............. 238
9.10 Lebesgue points for superharmonic functions ........... 244
9.11 Notes ................................................. 246
10 The Dirichlet problem for p-harmonic functions ............. 249
10.1 Continuous boundary values ............................ 250
10.2 The Kellogg property .................................. 251
10.3 Perron solutions ...................................... 253
10.4 Resolutivity of Newtonian functions ................... 255
10.5 Resolutivity of continuous functions .................. 261
10.6 Some consequences of resolutivity ..................... 263
10.7 Resolutivity of semicontinuous functions .............. 265
10.8 The /-harmonic measure ................................ 268
10.9 Poisson modification .................................. 271
10.10 The resolutivity problem ............................. 273
10.11 Notes ................................................ 275
11 Boundary regularity ........................................ 277
11.1 Barrier characterization of regular points ............ 277
11.2 Boundary regularity for the obstacle problem .......... 280
11.3 Characterizations of regularity ....................... 285
11.4 The Wiener criterion .................................. 288
11.5 Regularity componentwise .............................. 295
11.6 Fine continuity ....................................... 298
11.7 Notes ................................................. 301
12 Removable singularities .................................... 303
12.1 Removability .......................................... 303
12.2 Nonremovability ....................................... 309
12.3 Removable sets with positive capacity ................. 312
12.4 Nonunique removability ................................ 314
12.5 Notes ................................................. 316
13 Irregular boundary points .................................. 318
13.1 Semiregular and strongly irregular points ............. 318
13.2 Characterizations of semiregular points ............... 320
13.3 Characterizations of strongly irregular points ........ 325
13.4 The sets of semiregular and of strongly irregular
points ................................................ 327
13.5 Notes ................................................. 328
14 Regular sets and applications thereof ...................... 329
14.1 Regular sets .......................................... 329
14.2 Wiener solutions ...................................... 331
14.3 Classically superharmonic functions ................... 333
14.4 Notes ................................................. 334
Appendices
A Examples ................................................... 337
A.l N1,p in Euclidean spaces .............................. 337
A.2 Weighted Sobolev spaces on Rn ......................... 340
A.3 Uniform domains and power weights ..................... 348
A.4 Glueing spaces together ............................... 349
A.5 Graphs ................................................ 351
A.6 Carnot-Caratheodory spaces and Heisenberg groups ...... 354
A.7 Further examples ...................................... 357
A.8 Notes ................................................. 358
В Hajtasz-Sobolev and Cheeger-Sobolev spaces ................. 360
B.1 Hajiasz-Sobolev spaces ................................ 360
B.2 Cheeger-Sobolev spaces and differentiable structures .. 363
С Quasiminimizers ............................................ 365
Bibliography .................................................. 369
Index ......................................................... 389
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