Handbook of Differential Equations: Stationary Partial Differential Equations

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ISBN: 9780444511263 | 0444511261
Cover: Hardcover
Copyright: 7/6/2004

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The book could be a good companion for any graduate student in partial differential equations or in applied mathematics. Each chapter brings indeed new ideas and new techniques which can be used in these fields. The differents chapters can be read independently and are of great pedagogical value. The advanced researcher will find along the book the most recent achievements in various fields. · Independent chapters · Most recent advances in each fields · Hight didactic quality · Self contained · Excellence of the contributors · Wide range of topics

Preface

List of Contributors

vii

Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via Degree Theory

(70)

Catherine Bandle

Wolfgang Reichel

Degree theory

(14)

Introduction

(1)

Brouwer degree in finite dimensions

(3)

Leray-Schauder degree in Banach spaces

(3)

The index of an isolated solution

(2)

Asymptotically linear equations

(1)

Fixed point alternatives

(1)

Degree theory in unbounded domains

(1)

Degree theory in cones

(2)

Notes

(1)

Existence of solutions

(17)

Function spaces

(1)

Uniformly elliptic linear operators

(1)

Schauder estimates

(1)

Lp-estimates

(1)

Applications to boundary value problems

(2)

Comparison principles

(1)

Degree between sub- and supersolutions

(2)

Emden-Fowler type equations

(2)

Multiplicity results

(3)

Notes

(3)

Global continuation of solutions

(16)

A global implicit function theorem

(4)

Applications - continuation of solutions

(4)

Further applications

(6)

Notes

(2)

Bifurcation theory and related problems

(21)

Bifurcation from the trivial solution

(8)

Bifurcation from infinity

(7)

Perturbations at resonance

(3)

Acknowledgments

(1)

References

(3)

Stationary Navier-Stokes Problem in a Two-Dimensional Exterior Domain

(80)

Giovanni P. Galdi

Introduction

(3)

Outline of the chapter

(3)

Notation

(1)

Analysis of some linearized problems

(25)

The Stokes approximation

(7)

Some applications of Theorem 1.1

(4)

The Oseen approximation

(9)

The Oseen approximation in the limit of vanishing Reynolds number

101

(2)

A variant to the Oseen approximation

103

(2)

The nonlinear problem: Unique solvability for small Reynolds number and related results

105

(9)

Unique solvability at small Reynolds number

107

(4)

Limit of vanishing Reynolds number

111

(1)

Perturbation theory at finite Reynolds number

111

(3)

The nonlinear problem: On the solvability for arbitrary Reynolds number

114

(25)

Existence: Leray method

115

(2)

Existence: Fujita method

117

(2)

Some existence results when ξ = 0

119

(3)

On the pointwise asymptotic behavior of D-solutions

122

(11)

Asymptotic structure of D-solutions

133

(6)

The nonlinear problem: On the existence of symmetric solutions for arbitrary large Reynolds number

139

(12)

A remark about symmetric solutions

140

(1)

A key result

141

(8)

Existence of symmetric solutions for arbitrary large Reynolds number

149

(4)

Acknowledgments

153

(1)

References

154

Qualitative Properties of Solutions to Elliptic Problems

151

(84)

Wei-Ming Ni

Introduction

159

(4)

Concentrations of solutions: Single equations

163

(14)

Spike-layer solutions in elliptic boundary-value problems

163

(7)

Multi-peak spike-layer solutions in elliptic boundary-value problems

170

(3)

Solutions with multidimensional concentration sets

173

(4)

Remarks

177

(1)

Concentrations of solutions: Systems

177

(12)

The activator-inhibitor system

178

(3)

A Lotka-Volterra competition system with cross-diffusion

181

(4)

A chemotaxis system

185

(2)

Other systems

187

(2)

Stability of solutions

189

(15)

Single equations with Neumann boundary conditions

190

(4)

Single equations with Dirichlet boundary conditions

194

(2)

Shadow systems

196

(7)

Diffusion systems

203

(1)

Symmetry and related properties of solutions

204

(14)

Symmetry of semilinear elliptic equations in a ball

206

(2)

Symmetry of semilinear elliptic equations in an annulus

208

(1)

Symmetry of semilinear elliptic equations in entire space

208

(3)

Related monotonicity properties, level sets and more general domains

211

(3)

Generalizations and other types of equations

214

(1)

Symmetry of nonlinear elliptic systems

215

(2)

Miscellaneous results

217

(1)

Graphics and visualization

218

(17)

Mountain-pass and scaling algorithms

220

(2)

Visualization of solutions of singularly perturbed semilinear elliptic boundary value problems

222

(6)

Concluding remarks

228

(1)

Acknowledgment

228

(1)

References

228

(7)

On Some Basic Aspects of the Relationship between the Calculus of Variations and Differential Equations

235

(62)

Pablo Pedregal

Introduction

237

(5)

A little bit of history

242

(3)

The Euler-Lagrange equation: From VP to EL

245

(4)

Convexity: From EL to VP

249

(3)

Convexity: The direct method

252

(3)

Young measures

255

(5)

Scalar problems under pointwise constraints

260

(2)

Vector problems and systems of PDE

262

(7)

Vector problems and quasiconvexity

269

(3)

Second-order problems

272

(5)

Nonexistence: Lack of coercivity

277

(1)

Nonexistence: Lack of convexity

278

(4)

Generalized VP and generalized EL

282

(5)

Dynamical problems: Lack of convexity and lack of coercivity

287

(2)

Numerical approximation

289

(3)

Comments on other aspects of the CV

292

(5)

Acknowledgments

293

(1)

References

293

(4)

On a Class of Singular Perturbation Problems

297

(88)

Itai Shafrir

Introduction

299

(4)

Problems involving a double-well potential

303

(15)

BV spaces

303

(1)

Γ-convergence

304

(6)

Vector valued problems with a double-well potential

310

(5)

The Dirichlet boundary value problem

315

(3)

The work of Bethuel-Brezis-Helein

318

(20)

The case of zero degree boundary condition

321

(5)

The case of nonzero degree boundary condition

326

(12)

Minimization of Ginzburg-Landau energy when g is not S1-valued

338

(20)

The case of boundary condition without zeros

339

(5)

The case of boundary condition with zeros

344

(14)

The case of a general ``circular-well'' potential

358

(27)

A study of a degenerate metric

359

(7)

The singular perturbation problem

366

(15)

Acknowledgments

381

(1)

References

382

(3)

Nonlinear Spectral Problems for Degenerate Elliptic Operators

385

(106)

Peter Takac

Introduction

389

(3)

Preliminaries

392

(5)

Notation

392

(1)

A priori regularity results

393

(3)

Maximum and comparison principles

396

(1)

The first eigenvalue λ1

397

(5)

Convexity on the cone of positive functions

398

(2)

The inequality of Diaz and Saa

400

(1)

The first eigenfunction φ1

401

(1)

Subcritical spectral problems (λ < λ1)

402

(7)

Existence and uniqueness for λ < λ1

403

(3)

Nonexistence for λ = λ1

406

(1)

Anti-maximum principle for λ > λ1

407

(2)

Linearization about the first eigenfunction

409

(18)

Linearization and quadratization

410

(3)

The weighted Sobolev space Dπ1

413

(1)

A compact embedding with a weight for p >2

414

(3)

Simplicity of the first eigenvalue for the linearization

417

(5)

Another compact embedding for 1 < p <2

422

(1)

A few geometric inequalities

423

(4)

An improved Poincare inequality for p > 2

427

(9)

Statement and proof of Poincare's inequality

427

(6)

Fredholm alternative at λ1

433

(2)

Application to the embedding w01p → Lp

435

(1)

A saddle point method for p < 2

436

(5)

Simple saddle point geometry

437

(1)

A Palais--Smale condition

438

(2)

Fredholm alternative at λ1

440

(1)

Asymptotic behavior of large solutions

441

(20)

An approximation scheme

442

(3)

Convergence of approximate solutions

445

(1)

First-order estimates

446

(5)

Second-order estimates

451

(6)

A priori bounds

457

(3)

Nonexistence for λ = λ1

460

(1)

A variational approach

461

(14)

A minimax method

462

(1)

Asymptotic behavior of the constrained minima

463

(2)

Asymptotic behavior of jλ near ±∞

465

(3)

Existence of a solution for λ near λ1

468

(2)

Existence of two or three solutions

470

(5)

(Un)ordered pairs of sub-/supersolutions

475

(7)

Existence results using ordered pairs

476

(2)

Existence results using unordered pairs

478

(1)

(Un)ordered sets of solutions for λ = λ1

479

(3)

Bifurcations and the Fredholm alternative

482

(9)

An abstract global bifurcation result

483

(2)

Bifurcations from infinity

485

(2)

Acknowledgments

487

(1)

References

487

(4)

Analytical Aspects of Liouville-Type Equations with Singular Sources

491

(102)

Gabriella Tarantello

Introduction

493

(8)

Background material

501

(11)

The Liouville equation with singular sources in the plane

501

(4)

Analytic tools

505

(7)

Liouville-type equations: A concentration-compactness principle

512

(17)

Preliminaries

512

(4)

The blow-up technique

516

(7)

A concentration-compactness result

523

(6)

Quantization properties in the concentration phenomenon

529

(28)

Preliminaries

529

(1)

A version of Harnack's inequality

530

(7)

Inf + sup estimates

537

(7)

A quantization property

544

(8)

Examples

552

(5)

The effect of the boundary conditions on the blow-up analysis

557

(17)

Preliminaries

557

(2)

Pointwise estimates for the blow-up profile

559

(11)

The inf + sup estimate revised

570

(4)

Applications

574

(19)

The mean field equation over compact Riemannian surfaces

574

(6)

An existence result

580

(6)

Final comments

586

(1)

Acknowledgments

587

(1)

References

587

(6)

Elliptic Equations Involving Measures

593

(120)

Laurent Veron

Introduction

595

(3)

Linear equations

598

(15)

Elliptic equations in divergence form

598

(3)

The L1 framework

601

(4)

The measure framework

605

(3)

Representation theorems and boundary trace

608

(5)

Semilinear equations with absorption

613

(47)

The Marcinkiewicz spaces approach

615

(7)

Admissible measures and the Δ2-condition

622

(4)

The duality method

626

(8)

Removable singularities

634

(7)

Isolated singularities

641

(4)

The exponential and two-dimensional cases

645

(15)

Semilinear equations with source term

660

(20)

The basic approach

660

(2)

The convexity method

662

(10)

Semilinear equations with power source terms

672

(6)

Isolated singularities

678

(2)

Boundary singularities and boundary trace

680

(33)

Measures boundary data

680

(12)

Boundary singularities

692

(8)

The boundary trace problem

700

(6)

General nonlinearities

706

(3)

References

709

(4)

Author Index

713

(8)

Subject Index

721

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Handbook of Differential Equations: Stationary Partial Differential Equations

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