| Preface | p. v |
| General Technical Results | p. 1 |
| Introduction | p. 1 |
| Function Spaces | p. 1 |
| Regularity of Domains | p. 10 |
| Poincaré Inequality | p. 12 |
| Covering of Domains | p. 18 |
| Useful Techniques | p. 25 |
| Reverse Hölder's Inequality | p. 25 |
| Gehring's Result | p. 36 |
| Hole-Filling Technique of Widman | p. 38 |
| Inhomogeneous Hole-Filling | p. 40 |
| Green Function | p. 44 |
| Statement of Results | p. 44 |
| Proof of Theorem 1.26 | p. 45 |
| Estimates on log G | p. 46 |
| Estimates on Positive and Negative Powers of G | p. 49 |
| Harnack's Inequality | p. 52 |
| Proof of Theorem 1.27 | p. 57 |
| General Regularity Results | p. 63 |
| Introduction | p. 63 |
| Obtaining W1,p Regularity | p. 63 |
| Linear Equations | p. 63 |
| Nonlinear Problems | p. 66 |
| Obtaining C¿ Regularity | p. 70 |
| L∞ Bounds for Linear Problems | p. 70 |
| C¿ Regularity for Dirichlet Problems | p. 73 |
| C¿ Regularity for Linear Mixed Boundary Value Problems | p. 82 |
| C¿ Regularity in the Case n = 2 | p. 85 |
| Maximum Principle | p. 87 |
| Assumptions | p. 87 |
| Proof of Theorem 2.16 | p. 88 |
| More Regularity | p. 89 |
| From C¿ and <$>W^{1,p_0}<$>, p0 > 2, to <$>H_{{\rm loc}}^2<$> | p. 89 |
| Using the Linear Theory of Regularity | p. 96 |
| Full Regularity for a General Quasilinear Scalar Equation | p. 98 |
| Nonlinear Elliptic Systems Arising from Stochastic Games | p. 113 |
| Stochastic Games Background | p. 113 |
| Statement of the Problem and Results | p. 113 |
| Bellman Equations | p. 115 |
| Verification Property | p. 116 |
| Introduction to the Analytic Part | p. 118 |
| Estimates in Sobolev spaces and in C¿ | p. 120 |
| Assumptions and Statement of Results | p. 120 |
| Preliminaries | p. 122 |
| Proof of Theorem 3.7 | p. 125 |
| Estimates in L∞ | p. 127 |
| Assumptions | p. 127 |
| Statement of Results | p. 128 |
| Existence of Solutions | p. 129 |
| Setting of the Problem and Assumptions | p. 129 |
| Proof of Existence | p. 130 |
| Existence of a Weak Solution | p. 132 |
| Hamiltonians Arising from Games | p. 133 |
| Notation | p. 133 |
| Verification of the Assumptions for Hölder Regularity | p. 135 |
| Verification of the Assumptions for the L∞ Bound | p. 136 |
| The Case of Two Players with Different Coupling Terms in the Payoffs | p. 143 |
| Description of the Model and Statement of Results | p. 144 |
| L∞ Bounds | p. 145 |
| <$>H_0^1<$> Bound | p. 150 |
| Nonlinear Elliptic Systems Arising from Ergodic Control | p. 153 |
| Introduction | p. 153 |
| Assumptions and Statement of Results | p. 154 |
| Assumptions on the Hamiltonians | p. 154 |
| Statement of Results | p. 156 |
| Proof of Theorem 4.4 | p. 156 |
| First Estimates | p. 156 |
| Estimates on <$>u_{\epsilon}^\nu - \bar {u}_\epsilon^\nu<$> | p. 158 |
| End of Proof of Theorem 4.4 | p. 161 |
| Verification of the Assumptions | p. 162 |
| Notation | p. 162 |
| The Scalar Case | p. 163 |
| The General Case | p. 167 |
| A Variant of Theorem 4.4 | p. 169 |
| Statement of Results | p. 169 |
| Proof of Theorem 4.13 | p. 170 |
| Ergodic Problems in Rn | p. 175 |
| Presentation of the Problem | p. 175 |
| Existence Theorem for an Approximate Solution | p. 176 |
| Proof of Theorem 4.17 | p. 189 |
| Growth at Infinity | p. 191 |
| Uniqueness | p. 192 |
| Harmonic Mappings | p. 197 |
| Introduction | p. 197 |
| Extremals | p. 198 |
| Regularity | p. 200 |
| Hardy Spaces | p. 201 |
| Basic Properties | p. 201 |
| Main Regularity Result in the Hardy Space | p. 204 |
| Proof of Theorem 5.13 | p. 208 |
| Continuity when n = 2 | p. 208 |
| Proof of (5.35) and (5.36) | p. 216 |
| Proof of (5.37) | p. 218 |
| Atomic decomposition | p. 221 |
| Nonlinear Elliptic Systems Arising from the Theory of Semiconductors | p. 229 |
| Physical Background | p. 229 |
| Stationary Case Without Impact Ionization | p. 230 |
| Mathematical Setting | p. 230 |
| Proof of Theorem 6.1 | p. 233 |
| A Uniqueness Result | p. 240 |
| Local Regularity | p. 245 |
| Stationary Case with Impact Ionization | p. 246 |
| Setting of the Model | p. 246 |
| Proof of Theorem 6.5 | p. 248 |
| Impact Ionization Without Recombination | p. 257 |
| Statement of the Problem | p. 257 |
| Proof of Theorem 6.7 | p. 259 |
| Stationary Navier-Stokes Equations | p. 265 |
| Introduction | p. 265 |
| Regularity of "Maximum-Like Solutions" | p. 266 |
| Setting of the Problem | p. 266 |
| Some Regularity Properties of "Maximum-Like Solutions" | p. 267 |
| The Navier-Stokes Inequality | p. 273 |
| Hole-Filling | p. 275 |
| Full Regularity | p. 279 |
| Maximum Solutions and the NS Inequality | p. 280 |
| Notation and Setup | p. 280 |
| Proof of Theorem 7.8 | p. 281 |
| Existence of a Regular Solution for n ≤ 5 | p. 283 |
| Green Function Associated with Incompressible Flows | p. 283 |
| Approximation | p. 288 |
| Proof of Existence of a Maximum Solution for n ≤ 5 | p. 289 |
| Periodic Case: Existence of a Regular Solution for n < 10 | p. 291 |
| Approximation | p. 291 |
| A Specific Green Function | p. 292 |
| Main Results | p. 295 |
| Strongly Coupled Elliptic Systems | p. 299 |
| Introduction | p. 299 |
| <$>H_{{\rm loc}}^2<$> and Meyers's Regularity Results | p. 300 |
| Hölder Regularity | p. 305 |
| Preliminaries | p. 305 |
| Representation Using Spherical Functions | p. 308 |
| Statement of the Main Result | p. 311 |
| Additional Remarks | p. 317 |
| Hölder's Continuity up to the Boundary | p. 319 |
| C1+¿ Regularity | p. 329 |
| Auxiliary Inequalities | p. 329 |
| Main Result | p. 334 |
| Almost Everywhere Regularity | p. 338 |
| Regularity on Neighborhoods of Lebesgue Points | p. 338 |
| Proof of Theorem 8.22 | p. 339 |
| Regularity in the Uhlenbeck Case | p. 343 |
| Setting of the Problem | p. 343 |
| Proof of Theorem 8.24 | p. 344 |
| Counterexamples | p. 348 |
| Regularity for Mixed Boundary Value Systems | p. 352 |
| Stating the Problem | p. 352 |
| Proof of Theorem 8.25 | p. 354 |
| Proof of Lemma 8.28 | p. 359 |
| Further Regularity | p. 364 |
| Domain with a Corner. Mixed Boundary Conditions | p. 369 |
| Domain with a Corner. Dirichlet Boundary Conditions | p. 371 |
| Dual Approach to Nonlinear Elliptic Systems | p. 375 |
| Introduction | p. 375 |
| Preliminaries | p. 377 |
| Notation | p. 377 |
| Properties of the Operators ¿(u) and Du | p. 378 |
| Elasticity Models | p. 379 |
| Primal and Dual Problems | p. 379 |
| A Hybrid Model | p. 380 |
| <$>H_{{\rm loc}}^1<$> Theory for the Nonsymmetric Case | p. 381 |
| Presentation of the Problem | p. 381 |
| <$>H_{{\rm loc}}^1<$> Regularity | p. 382 |
| <$>H_{{\rm loc}}^1<$> Theory for the Symmetric Case | p. 391 |
| Presentation of the Problem | p. 391 |
| <$>H_{{\rm loc}}^1<$> Regularity | p. 391 |
| Reducing the Symmetric Case to the Nonsymmetric Case | p. 396 |
| <$>L_{{\rm loc}}^\infty<$> Theory for the Nonsymmetric Uhlenbeck Case | p. 398 |
| Setting of the Problem and Statement of Results | p. 398 |
| Proof of Theorem 9.8 | p. 399 |
| <$>W_{{\rm loc}}^{1,p}<$> Theory for the Nonsymmetric Case | p. 401 |
| Assumptions and Results | p. 401 |
| Proof of Theorem 9.9 | p. 402 |
| <$>C_{{\rm loc}}^{1+\delta}<$> Regularity for the Nonsymmetric Case | p. 405 |
| Setting of the Problem and Statement of Results | p. 405 |
| Preliminary Results | p. 406 |
| Proof of Theorem 9.10 | p. 410 |
| C¿ Regularity on Neighborhoods of Lebesgue Points for the Nonsymmetric Case | p. 413 |
| Setting of the Problem and Statement of Results | p. 413 |
| Proof of Theorem 9.11 | p. 414 |
| Additional Results in the Uhlenbeck Case | p. 418 |
| Nonlinear Elliptic Systems Arising from plasticity Theory | p. 421 |
| Introduction | p. 421 |
| Description of Models | p. 422 |
| Spaces U(¿), ¿(¿) | p. 422 |
| Hencky model | p. 423 |
| Norton-Hoff Model | p. 424 |
| Passing to the Limit | p. 426 |
| Estimates on the Displacement | p. 427 |
| The fj Derive from a Potential | p. 427 |
| Strict Interior Condition | p. 428 |
| Constituent Law for the Hencky model | p. 429 |
| <$>H_{{\rm loc}}^1<$> Regularity | p. 430 |
| Preliminaries | p. 430 |
| Uniform Estimates and Main Regularity Result | p. 432 |
| References | p. 435 |
| Index | p. 441 |
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