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Regularity Results for Nonlinear Elliptic Systems and Applications : Applied Mathematical Sciences - Alain Bensoussan

Regularity Results for Nonlinear Elliptic Systems and Applications

Applied Mathematical Sciences

Hardcover Published: 12th June 2002
ISBN: 9783540677567
Number Of Pages: 443

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The book collects many techniques that are helpul in obtaining regularity results for solutions of nonlinear systems of partial differential equations. They are then applied in various cases to provide useful examples and relevant results, particularly in fields like fluid mechanics, solid mechanics, semiconductor theory, or game theory. In general, these techniques are scattered in the journal literature and developed in the strict context of a given model. In the book, they are presented independently of specific models, so that the main ideas are explained, while remaining applicable to various situations. Such a presentation will facilitate application and implementation by researchers, as well as teaching to students.

From the reviews:

"The book under review presents several topics of the regularity theory for nonlinear elliptic equations and systems which have been developed in recent years. ... It may serve for teaching to higher graduate students and for researchers in the field of nonlinear PDE's." (Joachim Naumann, Zentralblatt MATH, Vol. 1055, 2005)

"It is a pleasure to review this new book by Bensoussan and Frehse; indeed this work covers remarkable, finely selected, updated material ... . this book is a very interesting and useful source of material for those interested in the latest developments of regularity theory of solutions of nonlinear elliptic systems. ... the book is an essential companion to the standard monographs already available and it is bound to become a standard and inevitable item in any list of references on the subject." (Giuseppe Mingione, Mathematical Reviews, 2004 a)

Prefacep. v
General Technical Resultsp. 1
Introductionp. 1
Function Spacesp. 1
Regularity of Domainsp. 10
Poincaré Inequalityp. 12
Covering of Domainsp. 18
Useful Techniquesp. 25
Reverse Hölder's Inequalityp. 25
Gehring's Resultp. 36
Hole-Filling Technique of Widmanp. 38
Inhomogeneous Hole-Fillingp. 40
Green Functionp. 44
Statement of Resultsp. 44
Proof of Theorem 1.26p. 45
Estimates on log Gp. 46
Estimates on Positive and Negative Powers of Gp. 49
Harnack's Inequalityp. 52
Proof of Theorem 1.27p. 57
General Regularity Resultsp. 63
Introductionp. 63
Obtaining W1,p Regularityp. 63
Linear Equationsp. 63
Nonlinear Problemsp. 66
Obtaining C¿ Regularityp. 70
L Bounds for Linear Problemsp. 70
C¿ Regularity for Dirichlet Problemsp. 73
C¿ Regularity for Linear Mixed Boundary Value Problemsp. 82
C¿ Regularity in the Case n = 2p. 85
Maximum Principlep. 87
Assumptionsp. 87
Proof of Theorem 2.16p. 88
More Regularityp. 89
From C¿ and <$>W^{1,p_0}<$>, p0 > 2, to <$>H_{{\rm loc}}^2<$>p. 89
Using the Linear Theory of Regularityp. 96
Full Regularity for a General Quasilinear Scalar Equationp. 98
Nonlinear Elliptic Systems Arising from Stochastic Gamesp. 113
Stochastic Games Backgroundp. 113
Statement of the Problem and Resultsp. 113
Bellman Equationsp. 115
Verification Propertyp. 116
Introduction to the Analytic Partp. 118
Estimates in Sobolev spaces and in C¿p. 120
Assumptions and Statement of Resultsp. 120
Preliminariesp. 122
Proof of Theorem 3.7p. 125
Estimates in Lp. 127
Assumptionsp. 127
Statement of Resultsp. 128
Existence of Solutionsp. 129
Setting of the Problem and Assumptionsp. 129
Proof of Existencep. 130
Existence of a Weak Solutionp. 132
Hamiltonians Arising from Gamesp. 133
Notationp. 133
Verification of the Assumptions for Hölder Regularityp. 135
Verification of the Assumptions for the L Boundp. 136
The Case of Two Players with Different Coupling Terms in the Payoffsp. 143
Description of the Model and Statement of Resultsp. 144
L Boundsp. 145
<$>H_0^1<$> Boundp. 150
Nonlinear Elliptic Systems Arising from Ergodic Controlp. 153
Introductionp. 153
Assumptions and Statement of Resultsp. 154
Assumptions on the Hamiltoniansp. 154
Statement of Resultsp. 156
Proof of Theorem 4.4p. 156
First Estimatesp. 156
Estimates on <$>u_{\epsilon}^\nu - \bar {u}_\epsilon^\nu<$>p. 158
End of Proof of Theorem 4.4p. 161
Verification of the Assumptionsp. 162
Notationp. 162
The Scalar Casep. 163
The General Casep. 167
A Variant of Theorem 4.4p. 169
Statement of Resultsp. 169
Proof of Theorem 4.13p. 170
Ergodic Problems in Rnp. 175
Presentation of the Problemp. 175
Existence Theorem for an Approximate Solutionp. 176
Proof of Theorem 4.17p. 189
Growth at Infinityp. 191
Uniquenessp. 192
Harmonic Mappingsp. 197
Introductionp. 197
Extremalsp. 198
Regularityp. 200
Hardy Spacesp. 201
Basic Propertiesp. 201
Main Regularity Result in the Hardy Spacep. 204
Proof of Theorem 5.13p. 208
Continuity when n = 2p. 208
Proof of (5.35) and (5.36)p. 216
Proof of (5.37)p. 218
Atomic decompositionp. 221
Nonlinear Elliptic Systems Arising from the Theory of Semiconductorsp. 229
Physical Backgroundp. 229
Stationary Case Without Impact Ionizationp. 230
Mathematical Settingp. 230
Proof of Theorem 6.1p. 233
A Uniqueness Resultp. 240
Local Regularityp. 245
Stationary Case with Impact Ionizationp. 246
Setting of the Modelp. 246
Proof of Theorem 6.5p. 248
Impact Ionization Without Recombinationp. 257
Statement of the Problemp. 257
Proof of Theorem 6.7p. 259
Stationary Navier-Stokes Equationsp. 265
Introductionp. 265
Regularity of "Maximum-Like Solutions"p. 266
Setting of the Problemp. 266
Some Regularity Properties of "Maximum-Like Solutions"p. 267
The Navier-Stokes Inequalityp. 273
Hole-Fillingp. 275
Full Regularityp. 279
Maximum Solutions and the NS Inequalityp. 280
Notation and Setupp. 280
Proof of Theorem 7.8p. 281
Existence of a Regular Solution for n ≤ 5p. 283
Green Function Associated with Incompressible Flowsp. 283
Approximationp. 288
Proof of Existence of a Maximum Solution for n ≤ 5p. 289
Periodic Case: Existence of a Regular Solution for n < 10p. 291
Approximationp. 291
A Specific Green Functionp. 292
Main Resultsp. 295
Strongly Coupled Elliptic Systemsp. 299
Introductionp. 299
<$>H_{{\rm loc}}^2<$> and Meyers's Regularity Resultsp. 300
Hölder Regularityp. 305
Preliminariesp. 305
Representation Using Spherical Functionsp. 308
Statement of the Main Resultp. 311
Additional Remarksp. 317
Hölder's Continuity up to the Boundaryp. 319
C1+¿ Regularityp. 329
Auxiliary Inequalitiesp. 329
Main Resultp. 334
Almost Everywhere Regularityp. 338
Regularity on Neighborhoods of Lebesgue Pointsp. 338
Proof of Theorem 8.22p. 339
Regularity in the Uhlenbeck Casep. 343
Setting of the Problemp. 343
Proof of Theorem 8.24p. 344
Counterexamplesp. 348
Regularity for Mixed Boundary Value Systemsp. 352
Stating the Problemp. 352
Proof of Theorem 8.25p. 354
Proof of Lemma 8.28p. 359
Further Regularityp. 364
Domain with a Corner. Mixed Boundary Conditionsp. 369
Domain with a Corner. Dirichlet Boundary Conditionsp. 371
Dual Approach to Nonlinear Elliptic Systemsp. 375
Introductionp. 375
Preliminariesp. 377
Notationp. 377
Properties of the Operators ¿(u) and Dup. 378
Elasticity Modelsp. 379
Primal and Dual Problemsp. 379
A Hybrid Modelp. 380
<$>H_{{\rm loc}}^1<$> Theory for the Nonsymmetric Casep. 381
Presentation of the Problemp. 381
<$>H_{{\rm loc}}^1<$> Regularityp. 382
<$>H_{{\rm loc}}^1<$> Theory for the Symmetric Casep. 391
Presentation of the Problemp. 391
<$>H_{{\rm loc}}^1<$> Regularityp. 391
Reducing the Symmetric Case to the Nonsymmetric Casep. 396
<$>L_{{\rm loc}}^\infty<$> Theory for the Nonsymmetric Uhlenbeck Casep. 398
Setting of the Problem and Statement of Resultsp. 398
Proof of Theorem 9.8p. 399
<$>W_{{\rm loc}}^{1,p}<$> Theory for the Nonsymmetric Casep. 401
Assumptions and Resultsp. 401
Proof of Theorem 9.9p. 402
<$>C_{{\rm loc}}^{1+\delta}<$> Regularity for the Nonsymmetric Casep. 405
Setting of the Problem and Statement of Resultsp. 405
Preliminary Resultsp. 406
Proof of Theorem 9.10p. 410
C¿ Regularity on Neighborhoods of Lebesgue Points for the Nonsymmetric Casep. 413
Setting of the Problem and Statement of Resultsp. 413
Proof of Theorem 9.11p. 414
Additional Results in the Uhlenbeck Casep. 418
Nonlinear Elliptic Systems Arising from plasticity Theoryp. 421
Introductionp. 421
Description of Modelsp. 422
Spaces U(¿), ¿(¿)p. 422
Hencky modelp. 423
Norton-Hoff Modelp. 424
Passing to the Limitp. 426
Estimates on the Displacementp. 427
The fj Derive from a Potentialp. 427
Strict Interior Conditionp. 428
Constituent Law for the Hencky modelp. 429
<$>H_{{\rm loc}}^1<$> Regularityp. 430
Preliminariesp. 430
Uniform Estimates and Main Regularity Resultp. 432
Referencesp. 435
Indexp. 441
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540677567
ISBN-10: 3540677569
Series: Applied Mathematical Sciences
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 443
Published: 12th June 2002
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 1.81