Applied Mathematical Sciences

By: Alain Bensoussan, Jens Frehse

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Hardcover | 12 June 2002

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460 Pages

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24.13 x 17.15 x 2.54

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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.

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"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)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Probability & Statistics Integral Calculus & Equations Medicine Medicine in General

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 W^1,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
C^1+¿ 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 f_j 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
Table of Contents provided by Publisher. All Rights Reserved.

Applied Mathematical Sciences

At a Glance

Hardcover

Industry Reviews

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