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1.Introduction 1

1.1.Partial differential equations 1

1.2.Examples 3

1.2.1.Single partial differential equations 3

1.2.2.Systems of partial differential equations 6

1.3.Strategies for studying PDE 6

1.3.1.Well-posed problems，classical solutions 7

1.3.2.Weak solutions 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 17

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 functions 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 solutions 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 solutions from envelopes 94

3.2.Characteristics 96

3.2.1.Derivation of characteristic ODE 96

3.2.2.Examples 99

3.2.3.Boundary conditions 102

3.2.4.Local solution 105

3.2.5.Applications 109

3.3.Introduction to Hamilton-Jacobi equations 114

3.3.1.Calculus of variations，Hamilton’s ODE 115

3.3.2.Legendre transform，Hopf-Lax formula 120

3.3.3.Weak solutions，uniqueness 128

3.4.Introduction to conservation laws 135

3.4.1.Shocks，entropy condition 136

3.4.2.Lax-Oleinik formula 143

3.4.3.Weak solutions，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 Solutions 167

4.1.Separation of variables 167

4.1.1.Examples 168

4.1.2.Application：Turing instability 172

4.2.Similarity solutions 176

4.2.1.Plane and traveling waves，solitons 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 functions 208

4.4.3.Hodograph and Legendre transforms 209

4.5.Asymptotics 211

4.5.1.Singular perturbations 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 analytic functions 237

4.6.3.Cauchy-Kovalevskaya Theorem 239

4.7.Problems 244

4.8.References 249

PARTⅡ：THEORY FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS 253

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 functions 264

5.3.2.Approximation by smooth functions 265

5.3.3.Global approximation by smooth functions 266

5.4.Extensions 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.Difierence quotients 291

5.8.3.Difierentiability a.e. 295

5.8.4.Hardy’s inequality 296

5.8.5.Fourier transform methods 297

5.9.Other spaces of functions 299

5.9.1.The space H-1 299

5.9.2.Spaces involving time 301

5.10.Problems 305

5.11.References 309

6.Second-Order Elliptic Equations 311

6.1.Definitions 311

6.1.1.Elliptic equations 311

6.1.2.Weak solutions 313

6.2.Existence of weak solutions 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

6.4.Maximum principles 344

6.4.1.Weak maximum principle 344

6.4.2.Strong maximum principle 347

6.4.3.Harnack’s inequality 351

6.5.Eigenvalues and eigenfunctions 354

6.5.1.Eigenvalues of symmetric elliptic operators 354

6.5.2.Eigenvalues of nonsymmetric elliptic operators 360

6.6.Problems 365

6.7.References 370

7.Linear Evolution Equations 371

7.1.Second-order parabolic equations 371

7.1.1.Definitions 372

7.1.2.Existence of weak solutions 375

7.1.3.Regularity 380

7.1.4.Maximum principles 389

7.2.Second-order hyperbolic equations 398

7.2.1.Definitions 398

7.2.2.Existence of weak solutions 401

7.2.3.Regularity 408

7.2.4.Propagation of disturbances 414

7.2.5.Equations in two variables 418

7.3.Hyperbolic systems of first-order equations 421

7.3.1.Definitions 421

7.3.2.Symmetric hyperbolic systems 423

7.3.3.Systems with constant coefficients 429

7.4.Semigroup theory 433

7.4.1.Definitions，elementary properties 434

7.4.2.Generating contraction semigroups 439

7.4.3.Applications 441

7.5.Problems 446

7.6.References 449

PARTⅢ：THEORY FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 453

8.The Calculus of Variations 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 solutions 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.Constraints 488

8.4.1.Nonlinear eigenvalue problems 488

8.4.2.Unilateral constraints，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 subsolutions and supersolutions 543

9.4.Nonexistence of solutions 547

9.4.1.Blow-up 547

9.4.2.Derrick-Pohozaev identity 551

9.5.Geometric properties of solutions 554

9.5.1.Star-shaped level sets 554

9.5.2.Radial symmetry 555

9.6.Gradient flows 560

9.6.1.Convex functions on Hilbert spaces 560

9.6.2.Subdifferentials and nonlinear semigroups 565

9.6.3.Applications 571

9.7.Problems 573

9.8.References 577

10.Hamilton-Jacobi Equations 579

10.1.Introduction，viscosity solutions 579

10.1.1.Definitions 581

10.1.2.Consistency 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 Conservation Laws 609

11.1.Introduction 609

11.1.1.Integral solutions 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 conservation laws 635

11.3.1.Riemann invariants 635

11.3.2.Nonexistence of smooth solutions 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 conservation laws 649

11.5.Problems 654

11.6.References 657

12.Nonlinear Wave Equations 659

12.1.Introduction 659

12.1.1.Conservation of energy 660

12.1.2.Finite propagation speed 660

12.2.Existence of solutions 663

12.2.1.Lipschitz nonlinearities 663

12.2.2.Short time existence 666

12.3.Semilinear wave equations 670

12.3.1.Sign conditions 670

12.3.2.Three space dimensions 674

12.3.3.Subcritical power nonlinearities 676

12.4.Critical power nonlinearity 679

12.5.Nonexistence of solutions 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 697

Appendix A：Notation 697

A.1.Notation for matrices 697

A.2.Geometric notation 698

A.3.Notation for functions 699

A.4.Vector-valued functions 703

A.5.Notation for estimates 703

A.6.Some comments about notation 704

Appendix B：Inequalities 705

B.1.Convex functions 705

B.2.Useful inequalities 706

Appendix C：Calculus 710

C.1.Boundaries 710

C.2.Gauss-Green Theorem 711

C.3.Polar coordinates，coarea formula 712

C.4.Moving regions 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.1.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.1.Lebesgue measure 729

E.2.Measurable functions and integration 730

E.3.Convergence theorems for integrals 731

E.4.Differentiation 732

E.5.Banach space-valued functions 733

Bibliography 735

Index 741