偏微分方程 第二版 英文版 埃文斯 高等教育出版社 Partial Differential Equati*** 2ed/Evans PDE研究生教材 pdf 夸克云 tct umd 下载 2025 azw3 kindle

这是当今关于偏微分方程 (PDE) 的最权威教材的第二版。它给出了PDE理论学习中现代技术的总览，特别注重非线性方程。本书内容广泛，阐述清晰，已经是PDE方面经典的研究生教材。在本版中，作者做了大量改动，包括：

新增非线性波动方程的一章，

超过 80 个新习题，

许多新的小节，

大大扩充了参考文献。

CONTENTS

Preface to second edition xvii

Preface to first edition xix

1. Introduction 1

1.1. Partial differential equati*** 1

1.2. Examples 3

1.2.1. Single partial differential equati*** 3

1.2.2. Systems of partial differential equati*** 6

1.3. Strategies for studying PDE 6

1.3.1. Well-posed problems, classical soluti*** 7

1.3.2. Weak soluti*** and regularity 7

1.3.3. Typical difficulties 9

1.4. Overview 9

1.5. Problems 12

1.6. References 13

PART I: REPRESENTATION FORMULAS

FOR SOLUTIONS

2. Four Important Linear PDE 17

2.1. Transport equation 18

2.1.1. Initial-value problem 18

2.1.2. Nonhomogeneous problem 19

2.2. Laplace's equation 20

2.2.1. Fundamental solution 21

2.2.2. Mean-value formulas 25

2.2.3. Properties of harmonic functi*** 26

2.2.4. Green's function 33

2.2.5. Energy methods 41

2.3. Heat equation 44

2.3.1. Fundamental solution 45

2.3.2. Mean-value formula 51

2.3.3. Properties of soluti*** 55

2.3.4. Energy methods 62

2.4. Wave equation 65

2.4.1. Solution by spherical means 67

2.4.2. Nonhomogeneous problem 80

2.4.3. Energy methods 82

2.5. Problems 84

2.6. References 90

3. Nonlinear First-Order PDE 91

3.1. Complete integrals, envelopes 92

3.1.1. Complete integrals 92

3.1.2. New soluti*** from envelopes 94

3.2. Characteristics 96

3.2.1. Derivation of characteristic ODE 96

3.2.2. Examples 99

3.2.3. Boundary conditi*** 102

3.2.4. Local solution 105

3.2.5. Applicati*** 109

3.3. Introduction to Hamilton-Jacobi equati*** 114

3.3.1. Calculus of variati***, Hamilton's ODE 115

3.3.2. Legendre transform, Hopf-Lax formula 120

3.3.3. Weak soluti***, uniqueness 128

3.4. Introduction to c***ervation laws 135

3.4.1. Shocks, entropy condition 136

3.4.2. Lax-Oleinik formula 143

3.4.3. Weak soluti***, uniqueness 148

3.4.4. Riemann's problem 153

3.4.5. Long time behavior 156

3.5. Problems 161

3.6. References 165

4. Other Ways to Represent Soluti*** 167

4.1. Separation of variables 167

4.1.1. Examples 168

4.1.2. Application: Turing instability 172

4.2. Similarity soluti*** 176

4.2.1. Plane and traveling waves, solit*** 176

4.2.2. Similarity under scaling 185

4.3. Transform methods 187

4.3.1. Fourier transform 187

4.3.2. Radon transform 196

4.3.3. Laplace transform 203

4.4. Converting nonlinear into linear PDE 206

4.4.1. Cole-Hopf transformation 206

4.4.2. Potential functi*** 208

4.4.3. Hodograph and Legendre transforms 209

4.5. Asymptotics 211

4.5.1. Singular perturbati*** 211

4.5.2. Laplace's method 216

4.5.3. Geometric optics, stationary phase 218

4.5.4. Homogenization 229

4.6. Power series 232

4.6.1. Noncharacteristic surfaces 232

4.6.2. Real ***ytic functi*** 237

4.6.3. Cauchy-Kovalevskaya Theorem 239

4.7. Problems 244

4.8. References 249

PART II: THEORY FOR LINEAR PARTIAL

DIFFERENTIAL EQUATIONS

5. Sobolev Spaces 253

5.1. Holder spaces 254

5.2. Sobolev spaces 255

5.2.1. Weak derivatives 255

5.2.2. Definition of Sobolev spaces 258

5.2.3. Elementary properties 261

5.3. Approximation 264

5.3.1. Interior approximation by smooth functi*** . . . 264

5.3.2. Approximation by smooth functi*** 265

5.3.3. Global approximation by smooth functi*** .... 266

5.4. Extensi*** 268

5.5. Traces 271

5.6. Sobolev inequalities 275

5.6.1. Gagliardo-Nirenberg-Sobolev inequality 276

5.6.2. Morrey's inequality 280

5.6.3. General Sobolev inequalities 284

5.7. Compactness 286

5.8. Additional topics 289

5.8.1. Poincare's inequalities 289

5.8.2. Difference quotients 291

5.8.3. Differentiability a.e 295

5.8.4. Hardy's inequality 296

5.8.5. Fourier transform methods 297

5.9. Other spaces of functi*** 299

5.9.1. The space Я"1 299

5.9.2. Spaces involving time 301

5.10. Problems 305

5.11. References 309

6. Second-Order Elliptic Equati*** 311

6.1. Definiti*** 311

6.1.1. Elliptic equati*** 311

6.1.2. Weak soluti*** 313

6.2. Existence of weak soluti*** 315

6.2.1. Lax-Milgram Theorem 315

6.2.2. Energy estimates 317

6.2.3. Fredholm alternative 320

6.3. Regularity 326

6.3.1. Interior regularity 327

6.3.2. Boundary regularity 334

***. Maximum principles 344

***.1. Weak maximum principle 344

***.2. Strong maximum principle 347

***.3. Harnack's inequality 351

6.5. Eigenvalues and eigenfuncti*** 354

6.5.1. Eigenvalues of symmetric elliptic operators .... 354

6.5.2. Eigenvalues of n***ymmetric elliptic operators . 360

6.6. Problems 365

6.7. References 370

7. Linear Evolution Equati*** 371

7.1. Second-order parabolic equati*** 371

7.1.1. Definiti*** 372

7.1.2. Existence of weak soluti*** 375

7.1.3. Regularity 380

7.1.4. Maximum principles 389

7.2. Second-order hyperbolic equati*** 398

7.2.1. Definiti*** 398

7.2.2. Existence of weak soluti*** 401

7.2.3. Regularity 408

7.2.4. Propagation of disturbances 414

7.2.5. Equati*** in two variables 418

7.3. Hyperbolic systems of first-order equati*** 421

7.3.1. Definiti*** 421

7.3.2. Symmetric hyperbolic systems 423

7.3.3. Systems with c***tant coefficients 429

7.4. Semigroup theory 433

7.4.1. Definiti***, elementary properties 434

7.4.2. Generating contraction semigroups 439

7.4.3. Applicati*** 441

7.5. Problems 446

7.6. References 449

PART III: THEORY FOR NONLINEAR PARTIAL

DIFFERENTIAL EQUATIONS

8. The Calculus of Variati*** 453

8.1. Introduction 453

8.1.1. Basic ideas 453

8.1.2. First variation, Euler-Lagrange equation 454

8.1.3. Second variation 458

8.1.4. Systems 459

8.2. Existence of minimizers 465

8.2.1. Coercivity, lower semicontinuity 465

8.2.2. Convexity 467

8.2.3. Weak soluti*** of Euler-Lagrange equation . . . 472

8.2.4. Systems 475

8.2.5. Local minimizers 480

8.3. Regularity 482

8.3.1. Second derivative estimates 483

8.3.2. Remarks on higher regularity 486

8.4. C***traints 488

8.4.1. Nonlinear eigenvalue problems 488

8.4.2. Unila***l c***traints, variational inequalities . 492

8.4.3. Harmonic maps 495

8.4.4. Incompressibility 497

8.5. Critical points 501

8.5.1. Mountain Pass Theorem 501

8.5.2. Application to semilinear elliptic PDE 507

8.6. Invariance, Noether's Theorem 511

8.6.1. Invariant variational problems 512

8.6.2. Noether's Theorem 513

8.7. Problems 520

8.8. References 525

9. Nonvariational Techniques 527

9.1. Monotonicity methods 527

9.2. Fixed point methods 533

9.2.1. Banach's Fixed Point Theorem 534

9.2.2. Schauder's, Schaefer's Fixed Point Theorems . . 538

9.3. Method of subsoluti*** and supersoluti*** 543

9.4. Nonexistence of soluti*** 547

9.4.1. Blow-up 547

9.4.2. Derrick-Pohozaev identity 551

9.5. Geometric properties of soluti*** 554

9.5.1. Star-shaped level sets 554

9.5.2. Radial symmetry 555

9.6. Gradient flows 560

9.6.1. Convex functi*** on Hilbert spaces 560

9.6.2. Subdifferentials and nonlinear semigroups .... 565

9.6.3. Applicati*** 571

9.7. Problems 573

9.8. References 577

10. Hamilton—Jacobi Equati*** 579

10.1. Introduction, viscosity soluti*** 579

10.1.1. Definiti*** 581

10.1.2. C***istency 583

10.2. Uniqueness 586

10.3. Control theory, dynamic programming 590

10.3.1. Introduction to optimal control theory 591

10.3.2. Dynamic programming 592

10.3.3. Hamilton-Jacobi-Bellman equation 594

10.3.4. Hopf-Lax formula revisited 600

10.4. Problems 603

10.5. References 606

11. Systems of C***ervation Laws 609

11.1. Introduction 609

11.1.1. Integral soluti*** 612

11.1.2. Traveling waves, hyperbolic systems 615

11.2. Riemann's problem 621

11.2.1. Simple waves 621

11.2.2. Rarefaction waves 624

11.2.3. Shock waves, contact discontinuities 625

11.2.4. Local solution of Riemann's problem 632

11.3. Systems of two c***ervation laws 635

11.3.1. Riemann invariants 635

11.3.2. Nonexistence of smooth soluti*** 639

11.4. Entropy criteria 641

11.4.1. Vanishing viscosity, traveling waves 642

11.4.2. Entropy/entropy-flux pairs 646

11.4.3. Uniqueness for scalar c***ervation laws 649

11.5. Problems 654

11.6. References 657

12. Nonlinear Wave Equati*** 659

12.1. Introduction 659

12.1.1. C***ervation of energy 660

12.1.2. Finite propagation speed 660

12.2. Existence of soluti*** 663

12.2.1. Lipschitz nonlinearities 663

12.2.2. Short time existence 666

12.3. Semilinea***ave equati*** 670

12.3.1. Sign conditi*** 670

12.3.2. Three space dimensi*** 674

12.3.3. Subcritical power nonlinearities 676

12.4. Critical power nonlinearity 679

12.5. Nonexistence of soluti*** 686

12.5.1. Nonexistence for negative energy 687

12.5.2. Nonexistence for small initial data 689

12.6. Problems 691

12.7. References 696

APPENDICES

Appendix A: Notation 697

A.l. Notation for matrices 697

A.2. Geometric notation 698

A.3. Notation for functi*** 699

A.4. Vector-valued functi*** 703

A.5. Notation for estimates 703

A.6. Some comments about notation 704

Appendix B: Inequalities 705

B.l. Convex functi*** 705

B.2. Useful inequalities 706

Appendix C: Calculus 710

C.l. Boundaries 710

C.2. Gauss-Green Theorem 711

C.3. Polar coordinates, coarea formula 712

C.4. Moving regi*** 713

C.5. Convolution and smoothing 713

C.6. Inverse Function Theorem 716

C.7. Implicit Function Theorem 717

C.8. Uniform convergence 718

Appendix D: Functional Analysis 719

D.l. Banach spaces 719

D.2. Hilbert spaces 720

D.3. Bounded linear operators 721

D.4. Weak convergence 723

D.5. Compact operators, Fredholm theory 724

D.6. Symmetric operators 728

Appendix E: Measure Theory 729

E.l. Lebesgue measure 729

E.2. Measurable functi*** and integration 730

E.3. Convergence theorems for integrals 731

E.4. Differentiation 732

E.5. Banach space-valued functi*** 733

Bibliography 735

Index 741

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a. PDE theory is (mostly) not restricted to two independent variables.

b. Many interesting equati*** are nonlinear.

c. Understanding generalized soluti*** is fundamental.

d. PDE theory is not a branch of function ***ysis.

e. Notation is a nightmare.

f. Good theory is (almost) as useful as exact formulas.

书籍介绍

这是当今关于偏微分方程 (PDE) 的最权威教材的第二版。它给出了PDE理论学习中现代技术的总览，特别注重非线性方程。本书内容广泛，阐述清晰，已经是PDE方面经典的研究生教材。在本版中，作者做了大量改动，包括：

新增非线性波动方程的一章，

超过 80 个新习题，

许多新的小节，

大大扩充了参考文献。