Handbook of Differential Equations: Stationary Partial Differential Equations

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ISBN: 9780444528469 | 0444528466
Cover: Hardcover
Copyright: 10/16/2006

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This handbook is volume III in a series devoted to stationary partial differential quations. Similarly as volumes I and II, it is a collection of self contained state-of-the-art surveys written by well known experts in the field. The topics covered by this handbook include singular and higher order equations, problems near critically, problems with anisotropic nonlinearities, dam problem, T-convergence and Schauder-type estimates. These surveys will be useful for both beginners and experts and speed up the progress of corresponding (rapidly developing and fascinating) areas of mathematics. Key features: - Written by well-known experts in the field - Self-contained volume in series covering one of the most rapid developing topics in mathematics - Written by well-known experts in the field - Self-contained volume in series covering one of the most rapid developing topics in mathematics

Preface

List of Contributors

vii

Contents of Volume I

Contents of Volume II

xiii

1. Elliptic Equations with Anisotropic Nonlinearity and Nonstandard Growth Conditions

(100)

S. Antontsev and S. Shmarev

1. Introduction

(9)

1.1. Assumptions and results

(3)

1.2. Physical motivation

(4)

1.3. Previous work

(2)

2. The Lebesgue and Sobolev spaces with variable exponents

(4)

2.1. Spaces Lp(x)(Ω) and W10,p(x)(Ω)

(2)

2.2. Anisotropic spaces

(1)

2.3. Embedding theorems

(1)

3. Existence theorems

(18)

3.1. Generalized p(x)-Laplace equation

(8)

3.2. Generalized diffusion equation

(8)

3.3. Equations with convection terms

(2)

4. Uniqueness theorems

(9)

4.1. Uniqueness of solution of the generalized p(x)-Laplace equation

(4)

4.2. Uniqueness of solution of the generalized diffusion equation

(3)

5. Localization caused by the diffusion—absorption balance

(13)

5.1. Generalized p(x)-Laplace equation

(4)

5.2. The energy relation

(2)

5.3. The ordinary differential inequality

(6)

5.4. Equations with convection terms

(1)

6. Directional localization caused by anisotropic diffusion

(16)

6.1. Generalized diffusion equation

(12)

6.2. Generalized p(x)-Laplace equation

(4)

7. Problems on unbounded domains

(4)

7.1. Generalized diffusion equation

(4)

7.2. Generalized p(x)-Laplace equation

(1)

8. Systems of elliptic equations

(6)

8.1. Existence of solutions

(1)

8.2. Localization properties

(3)

8.3. Systems of other types

(1)

9. Examples: localization in borderline cases

(15)

9.1. Illustrative examples

(13)

9.2. The ordinary differential inequality in the limit case

(2)

Acknowledgements

(1)

References

(6)

Preface

103

2. A Handbook of Γ-Convergence

101

(114)

A. Braides

1. Introduction

103

(1)

Notation

109

(1)

2. General theory of Γ-convergence

110

(7)

2.1. The definitions of Γ-convergence

110

(3)

2.2. Γ-convergence and lower semicontinuity

113

(1)

2.3. Computation of Γ-limits

114

(1)

2.4. Properties of Γ-convergence

115

(2)

2.5. Development by Γ-convergence

117

(1)

3. Localization methods

117

(7)

3.1. Supremum of measures

118

(1)

3.2. The blow-up technique

119

(1)

3.3. A general compactness procedure

120

(2)

3.4. The "slicing" method

122

(2)

4. Local integral functionals on Sobolev spaces

124

(10)

4.1. A prototypical compactness theorem

124

(4)

4.2. Useful technical results

128

(3)

4.3. Convergence of quadratic forms

131

(1)

4.4. Degenerate limits

132

(2)

5. Homogenization of integral functionals

134

(10)

5.1. The asymptotic homogenization formula

135

(2)

5.2. The convex case: the cell-problem formula

137

(1)

5.3. Homogenization of quadratic forms

138

(2)

5.4. Bounds on composites

140

(2)

5.5. Homogenization of metrics

142

(2)

6. Perforated domains and relaxed Dirichlet problems

144

(11)

6.1. Dirichlet boundary conditions: a direct approach

144

(5)

6.2. Relaxed Dirichlet problems

149

(2)

6.3. Neumann boundary conditions: an extension lemma

151

(2)

6.4. Double-porosity homogenization

153

(2)

7. Phase-transition problems

155

(18)

7.1. Interfacial energies

155

(3)

7.2. Gradient theory of phase transitions

158

(6)

7.3. A compactness result

164

(1)

7.4. Other functionals generating phase-transitions

165

(6)

7.5. Some extensions

171

(2)

8. Concentration problems

173

(7)

8.1. Ginzburg–Landau

173

(3)

8.2. Critical-growth problems

176

(4)

9. Dimension-reduction problems

180

(132)

9.1. The Le Dret–Raoult result

180

(3)

9.2. A compactness theorem

183

(2)

9.3. Higher-order Γ-limits

185

(1)

10. Approximation of free-discontinuity problems

186

(13)

10.1. Special functions with bounded variation

187

(3)

10.2. The Ambrosio–Tortorelli approximation

190

(3)

10.3. Other approximations

193

(3)

10.4. Approximation of curvature functionals

196

(3)

11. Continuum limits of lattice systems

199

(9)

11.1. Continuum energies on Sobolev spaces

200

(5)

11.2. Continuum energies on discontinuous functions

205

(3)

Note to the bibliography

208

(1)

References

208

(7)

3. Bubbling in Nonlinear Elliptic Problems Near Criticality

215

(102)

M. del Pino and M. Musso

1. Introduction

217

(8)

2. Nearly critical bubbling: the proof of Theorem 1.1

225

(30)

2.1. Ansatz and scheme of the proof

225

(8)

2.2. Variational reduction and conclusion of the proof

233

(22)

3. Solvability of slightly supercritical problems and the topology of the domain

255

(14)

3.1. The case of a small hole

255

(11)

3.2. Bubbling under symmetries

266

(3)

4. The Brezis–Nirenberg problem in dimension N = 3: the proof of Theorem 1.2

269

(16)

4.1. Energy expansion of single bubbling

270

(3)

4.2. The method of proof

273

(3)

4.3. The linear problem

276

(1)

4.4. Solving the nonlinear problem

277

(1)

4.5. Variational formulation of the reduced problem for k = 1

278

(1)

4.6. Proof of Theorem 1.2, part (a): single bubbling

279

(1)

4.7. Proof of Theorem 1.2, part (b): multiple bubbling

280

(5)

5. Liouville-type equations

285

(27)

5.1. Proof of Theorem 1.3

285

(24)

5.2. A related 2-d problem involving nonlinearity with large exponent

309

(3)

Acknowledgement

312

(1)

References

312

(5)

4. Singular Elliptic and Parabolic Equations

317

(84)

J. Hernandez and F.J. Mancebo

1. Introduction

319

(6)

2. The spectrum of the linear eigenvalue problem

325

(8)

3. Existence theorems in the general case α > 0

333

(4)

4. Existence of solutions via sub- and supersolutions

337

(3)

5. Uniqueness of positive solutions for sublinear problems

340

(1)

6. Regularity and boundary behavior

341

(9)

7. Differentiability for some singular nonlinear problems

350

(3)

8. The associated parabolic problem: linearized stability

353

(5)

9. Stabilization

358

(4)

10. Applications

362

(8)

10.1. Power law: Lu = λK(x)uq

362

(2)

10.2. First combination of two power laws: Lu + M (x)uP = λK(x)uq

364

(3)

10.3. Second combination of power laws: Lu = λK(x)uq + M(x)up

367

(3)

10.4. Third combination of power laws: Lu + M(x)up = λK x)uq , with M changing sign

370

(1)

11. Variational methods: some multiplicity results

370

(6)

12. Results for the radial and the one-dimensional problem

376

(7)

13. Free boundary solutions: existence and properties

383

(5)

Acknowledgements

388

(1)

Appendix A. Proof of Proposition 2.3

389

(4)

Appendix B. A strong maximum principle for second-order equations with locally bounded coefficients

393

(2)

References

395

(6)

5. Schauder-Type Estimates and Applications

401

(64)

S. Kichenassamy

1. Introduction

403

(8)

1.1. What are Schauder-type estimates?

403

(1)

1.2. Why do we need Schauder estimates?

404

(1)

1.3. Why so many methods of proof?

405

(1)

1.4. Classification of proofs

406

(1)

1.5. Generalizations and variants

407

(1)

1.6. What process were the Schauder estimates discovered by?

408

(2)

1.7. Outline of the chapter

410

(1)

2. Holder spaces

411

(8)

2.1. First definitions

411

(1)

2.2. Dyadic decomposition

412

(1)

2.3. Weighted norms

413

(2)

2.4. Interpolation inequalities

415

(1)

2.5. Properties of the distance function

416

(2)

2.6. Integral characterization of Wilder continuity

418

(1)

3. Interior estimates for the Laplacian

419

(13)

3.1. Direct arguments from potential theory

419

(4)

3.2. C1+α estimates via the maximum principle

423

(2)

3.3. C2+α estimates via Littlewood–Paley theory

425

(1)

3.4. Variational approach

426

(4)

3.5. Other methods

430

(2)

4. Perturbation of coefficients

432

(3)

4.1. Basic estimate

432

(2)

4.2. Estimates up to the boundary

434

(1)

5. Fuchsian operators on C2+α domains

435

(5)

5.1. First "type (I)" result

436

(2)

5.2. Second "type (I)" result

438

(2)

6. Applications

440

(21)

6.1. Method of continuity

440

(1)

6.2. Basic fixed-point theorems for compact operators

441

(4)

6.3. Fixed-point theory and the Dirichlet problem

445

(1)

6.4. Eigenfunctions and applications

446

(2)

6.5. Method of sub- and supersolutions

448

(1)

6.6. Asymptotics near isolated singularities or at infinity

449

(2)

6.7. Asymptotics for boundary blow-up

451

(3)

6.8. First comparison argument

454

(7)

References

461

(4)

6. The Dam Problem

465

(88)

A. Lyaghfouri

Introduction

467

(4)

The variational inequalities approach by Baiocchi

467

(1)

The weak formulation of Alt and Brezis–Kinderlehrer–Stampacchia

468

(1)

Outline of the chapter

469

(1)

Notation

470

(1)

1. A unified formulation of the dam problem

471

(20)

1.1. Formulation of the problem

471

(4)

1.2. Existence of a solution

475

(11)

1.3. Regularity and monotonicity of the solutions

486

(5)

2. The dam problem with Dirichlet boundary condition

491

(25)

2.1. Some properties of the solutions

491

(4)

2.2. Continuity of the free boundary

495

(3)

2.3. Existence and uniqueness of minimal and maximal solutions

498

(9)

2.4. Reservoirs-connected solution

507

(3)

2.5. Uniqueness of the reservoirs-connected solution

510

(6)

3. The dam problem with leaky boundary condition

516

(35)

3.1. Properties of the solutions

516

(10)

3.2. Continuity of the free boundary

526

(7)

3.3. Existence and uniqueness of minimal and maximal solutions

533

(11)

3.4. Reservoirs-connected solution

544

(1)

3.5. Uniqueness of the reservoirs-connected solution

545

(6)

Acknowledgement

551

(1)

References

551

(2)

7. Nonlinear Eigenvalue Problems for Higher-Order Model Equations

553

(52)

L.A. Peletier

1. Introduction

555

(5)

2. Decreasing nonlinearity ƒ(u)

560

(13)

2.1. Existence of a periodic solution

561

(4)

2.2. Uniqueness

565

(4)

2.3. Asymptotics

569

(4)

3. A superlinear bifurcation problem

573

(12)

3.1. Preliminaries

574

(2)

3.2. Multibump periodic solutions

576

(7)

3.3. Branches of periodic solutions

583

(2)

4. A sublinear bifurcation problem

585

(8)

5. A multiscale analysis

593

(10)

5.1. Multibump solutions

594

(5)

5.2. Branches of multibump periodic solutions

599

(4)

Acknowledgements

603

(1)

References

603

(2)

Author Index

605

(8)

Subject Index

613

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Handbook of Differential Equations: Stationary Partial Differential Equations

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