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Regularity Results for Nonlinear Elliptic Systems and Applications

Applied Mathematical Sciences

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

Hardcover

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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)

 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 Table of Contents provided by Publisher. 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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