Partial differential equations Second Edition volume 19PDF|Epub|txt|kindle电子书版本网盘下载

Partial differential equations Second Edition volume 19PDF格式电子书版下载

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1.Introduction1

1.1.Partial differential equations1

1.2.Examples3

1.2.1.Single partial differential equations3

1.2.2.Systems of partial differential equations6

1.3.Strategies for studying PDE6

1.3.1.Well-posed problems，classical solutions7

1.3.2.Weak solutions and regularity7

1.3.3.Typical difficulties9

1.4.Overview9

1.5.Problems12

1.6.References13

PART Ⅰ：REPRESENTATION FORMULAS FOR SOLUTIONS17

2.Four Important Linear PDE17

2.1.Transport equation18

2.1.1.Initial-value problem18

2.1.2.Nonhomogeneous problem19

2.2.Laplace’s equation20

2.2.1.Fundamental solution21

2.2.2.Mean-value formulas25

2.2.3.Properties of harmonic functions26

2.2.4.Green’s function33

2.2.5.Energy methods41

2.3.Heat equation44

2.3.1.Fundamental solution45

2.3.2.Mean-value formula51

2.3.3.Properties of solutions55

2.3.4.Energy methods62

2.4.Wave equation65

2.4.1.Solution by spherical means67

2.4.2.Nonhomogeneous problem80

2.4.3.Energy methods82

2.5.Problems84

2.6.References90

3.Nonlinear First-Order PDE91

3.1.Complete integrals，envelopes92

3.1.1.Complete integrals92

3.1.2.New solutions from envelopes94

3.2.Characteristics96

3.2.1.Derivation of characteristic ODE96

3.2.2.Examples99

3.2.3.Boundary conditions102

3.2.4.Local solution105

3.2.5.Applications109

3.3.Introduction to Hamilton-Jacobi equations114

3.3.1.Calculus of variations，Hamilton’s ODE115

3.3.2.Legendre transform，Hopf-Lax formula120

3.3.3.Weak solutions，uniqueness128

3.4.Introduction to conservation laws135

3.4.1.Shocks，entropy condition136

3.4.2.Lax-Oleinik formula143

3.4.3.Weak solutions，uniqueness148

3.4.4.Riemann’s problem153

3.4.5.Long time behavior156

3.5.Problems161

3.6.References165

4.Other Ways to Represent Solutions167

4.1.Separation of variables167

4.1.1.Examples168

4.1.2.Application：Turing instability172

4.2.Similarity solutions176

4.2.1.Plane and traveling waves，solitons176

4.2.2.Similarity under scaling185

4.3.Transform methods187

4.3.1.Fourier transform187

4.3.2.Radon transform196

4.3.3.Laplace transform203

4.4.Converting nonlinear into linear PDE206

4.4.1.Cole-Hopf transformation206

4.4.2.Potential functions208

4.4.3.Hodograph and Legendre transforms209

4.5.Asymptotics211

4.5.1.Singular perturbations211

4.5.2.Laplace’s method216

4.5.3.Geometric optics，stationary phase218

4.5.4.Homogenization229

4.6.Power series232

4.6.1.Noncharacteristic surfaces232

4.6.2.Real analytic functions237

4.6.3.Cauchy-Kovalevskaya Theorem239

4.7.Problems244

4.8.References249

PARTⅡ：THEORY FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS253

5.Sobolev Spaces253

5.1.Holder spaces254

5.2.Sobolev spaces255

5.2.1.Weak derivatives255

5.2.2.Definition of Sobolev spaces258

5.2.3.Elementary properties261

5.3.Approximation264

5.3.1.Interior approximation by smooth functions264

5.3.2.Approximation by smooth functions265

5.3.3.Global approximation by smooth functions266

5.4.Extensions268

5.5.Traces271

5.6.Sobolev inequalities275

5.6.1.Gagliardo-Nirenberg-Sobolev inequality276

5.6.2.Morrey’s inequality280

5.6.3.General Sobolev inequalities284

5.7.Compactness286

5.8.Additional topics289

5.8.1.Poincare’s inequalities289

5.8.2.Difierence quotients291

5.8.3.Difierentiability a.e.295

5.8.4.Hardy’s inequality296

5.8.5.Fourier transform methods297

5.9.Other spaces of functions299

5.9.1.The space H-1299

5.9.2.Spaces involving time301

5.10.Problems305

5.11.References309

6.Second-Order Elliptic Equations311

6.1.Definitions311

6.1.1.Elliptic equations311

6.1.2.Weak solutions313

6.2.Existence of weak solutions315

6.2.1.Lax-Milgram Theorem315

6.2.2.Energy estimates317

6.2.3.Fredholm alternative320

6.3.Regularity326

6.3.1.Interior regularity327

6.3.2.Boundary regularity334

6.4.Maximum principles344

6.4.1.Weak maximum principle344

6.4.2.Strong maximum principle347

6.4.3.Harnack’s inequality351

6.5.Eigenvalues and eigenfunctions354

6.5.1.Eigenvalues of symmetric elliptic operators354

6.5.2.Eigenvalues of nonsymmetric elliptic operators360

6.6.Problems365

6.7.References370

7.Linear Evolution Equations371

7.1.Second-order parabolic equations371

7.1.1.Definitions372

7.1.2.Existence of weak solutions375

7.1.3.Regularity380

7.1.4.Maximum principles389

7.2.Second-order hyperbolic equations398

7.2.1.Definitions398

7.2.2.Existence of weak solutions401

7.2.3.Regularity408

7.2.4.Propagation of disturbances414

7.2.5.Equations in two variables418

7.3.Hyperbolic systems of first-order equations421

7.3.1.Definitions421

7.3.2.Symmetric hyperbolic systems423

7.3.3.Systems with constant coefficients429

7.4.Semigroup theory433

7.4.1.Definitions，elementary properties434

7.4.2.Generating contraction semigroups439

7.4.3.Applications441

7.5.Problems446

7.6.References449

PARTⅢ：THEORY FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS453

8.The Calculus of Variations453

8.1.Introduction453

8.1.1.Basic ideas453

8.1.2.First variation，Euler-Lagrange equation454

8.1.3.Second variation458

8.1.4.Systems459

8.2.Existence of minimizers465

8.2.1.Coercivity，lower semicontinuity465

8.2.2.Convexity467

8.2.3.Weak solutions of Euler-Lagrange equation472

8.2.4.Systems475

8.2.5.Local minimizers480

8.3.Regularity482

8.3.1.Second derivative estimates483

8.3.2.Remarks on higher regularity486

8.4.Constraints488

8.4.1.Nonlinear eigenvalue problems488

8.4.2.Unilateral constraints，variational inequalities492

8.4.3.Harmonic maps495

8.4.4.Incompressibility497

8.5.Critical points501

8.5.1.Mountain Pass Theorem501

8.5.2.Application to semilinear elliptic PDE507

8.6.Invariance，Noether’s Theorem511

8.6.1.Invariant variational problems512

8.6.2.Noether’s Theorem513

8.7.Problems520

8.8.References525

9.Nonvariational Techniques527

9.1.Monotonicity methods527

9.2.Fixed point methods533

9.2.1.Banach’s Fixed Point Theorem534

9.2.2.Schauder’s，Schaefer’s Fixed Point Theorems538

9.3.Method of subsolutions and supersolutions543

9.4.Nonexistence of solutions547

9.4.1.Blow-up547

9.4.2.Derrick-Pohozaev identity551

9.5.Geometric properties of solutions554

9.5.1.Star-shaped level sets554

9.5.2.Radial symmetry555

9.6.Gradient flows560

9.6.1.Convex functions on Hilbert spaces560

9.6.2.Subdifferentials and nonlinear semigroups565

9.6.3.Applications571

9.7.Problems573

9.8.References577

10.Hamilton-Jacobi Equations579

10.1.Introduction，viscosity solutions579

10.1.1.Definitions581

10.1.2.Consistency583

10.2.Uniqueness586

10.3.Control theory，dynamic programming590

10.3.1.Introduction to optimal control theory591

10.3.2.Dynamic programming592

10.3.3.Hamilton-Jacobi-Bellman equation594

10.3.4.Hopf-Lax formula revisited600

10.4.Problems603

10.5.References606

11.Systems of Conservation Laws609

11.1.Introduction609

11.1.1.Integral solutions612

11.1.2.Traveling waves，hyperbolic systems615

11.2.Riemann’s problem621

11.2.1.Simple waves621

11.2.2.Rarefaction waves624

11.2.3.Shock waves，contact discontinuities625

11.2.4.Local solution of Riemann’s problem632

11.3.Systems of two conservation laws635

11.3.1.Riemann invariants635

11.3.2.Nonexistence of smooth solutions639

11.4.Entropy criteria641

11.4.1.Vanishing viscosity，traveling waves642

11.4.2.Entropy/entropy-flux pairs646

11.4.3.Uniqueness for scalar conservation laws649

11.5.Problems654

11.6.References657

12.Nonlinear Wave Equations659

12.1.Introduction659

12.1.1.Conservation of energy660

12.1.2.Finite propagation speed660

12.2.Existence of solutions663

12.2.1.Lipschitz nonlinearities663

12.2.2.Short time existence666

12.3.Semilinear wave equations670

12.3.1.Sign conditions670

12.3.2.Three space dimensions674

12.3.3.Subcritical power nonlinearities676

12.4.Critical power nonlinearity679

12.5.Nonexistence of solutions686

12.5.1.Nonexistence for negative energy687

12.5.2.Nonexistence for small initial data689

12.6.Problems691

12.7.References696

APPENDICES697

Appendix A：Notation697

A.1.Notation for matrices697

A.2.Geometric notation698

A.3.Notation for functions699

A.4.Vector-valued functions703

A.5.Notation for estimates703

A.6.Some comments about notation704

Appendix B：Inequalities705

B.1.Convex functions705

B.2.Useful inequalities706

Appendix C：Calculus710

C.1.Boundaries710

C.2.Gauss-Green Theorem711

C.3.Polar coordinates，coarea formula712

C.4.Moving regions713

C.5.Convolution and smoothing713

C.6.Inverse Function Theorem716

C.7.Implicit Function Theorem717

C.8.Uniform convergence718

Appendix D：Functional Analysis719

D.1.Banach spaces719

D.2.Hilbert spaces720

D.3.Bounded linear operators721

D.4.Weak convergence723

D.5.Compact operators，Fredholm theory724

D.6.Symmetric operators728

Appendix E：Measure Theory729

E.1.Lebesgue measure729

E.2.Measurable functions and integration730

E.3.Convergence theorems for integrals731

E.4.Differentiation732

E.5.Banach space-valued functions733

Bibliography735

Index741