偏微分方程 第1卷 第2版 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

偏微分方程 第1卷 第2版 英文PDF格式电子书版下载

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1 Basic Theory of ODE and Vector Fields1

1 The derivative3

2 Fundamental local existence theorem for ODE9

3 Inverse function and implicit function theorems12

4 Constant-coefficient linear systems;exponentiation of matrices16

5 Variable-coefficient linear systems of ODE：Duhamel's principle26

6 Dependence of solutions on initial data and on other parameters31

7 Flows and vector fields35

8 Lie brackets40

9 Commuting flows;Frobenius's theorem43

10 Hamiltonian systems47

11 Geodesics51

12 Variational problems and the stationary action principle59

13 Differential forms70

14 The symplectic form and canonical transformations83

15 First-order,scalar,nonlinear PDE89

16 Completely integrable hamiltonian systems96

17 Examples of integrable systems;central force problems101

18 Relativistic motion105

19 Topological applications of differential forms110

20 Critical points and index of a vector field118

A Nonsmooth vector fields122

References125

2 The Laplace Equation and Wave Equation127

1 Vibrating strings and membranes129

2 The divergence of a vector field140

3 The covariant derivative and divergence of tensor fields145

4 The Laplace operator on a Riemannian manifold153

5 The wave equation on a product manifold and energy conservation156

6 Uniqueness and finitepropagation speed162

7 Lorentz manifolds and stress-energy tensors166

8 More general hyperbolic equations;energy estimates172

9 The symbol of a differential operator and a general Green-Stokes formula176

10 The Hodge Laplacian on k-forms180

11 Maxwell's equations184

References194

3 Fourier Analysis,Distributions,and Constant-Coefficient LinearPDE197

1 Fourier series198

2 Harmonic functions and holomorphic functions in the plane209

3 The Fourier transform222

4 Distributions and tempered distributions230

5 The classical evolution equations244

6 Radial distributions,polar coordinates,and Bessel functions263

7 The method ofimages and Poisson's summation formula273

8 Homogeneous distributions and principal value distributions278

9 Elliptic operators286

10 Local solvability of constant-coefficient PDE289

11 The discrete Fourier transform292

12 The fast Fourier transform301

A The mighty Gaussian and the sublime gamma function306

References312

4 Sobolev Spaces315

1 Sobolev spaces on Rn315

2 The complex interpolation method321

3 Sobolev spaces on compact manifolds328

4 Sobolev spaces on bounded domains331

5 The Sobolev spaces Hs 0（Ω）338

6 The Schwartz kernel theorem345

7 Sobolev spaces on rough domains349

References351

5 Linear Elliptic Equations353

1 Existence and regularity of solutions to the Dirichlet problem354

2 The weak and strong maximum principles364

3 The Dirichlet problem on the ball in Rn373

4 The Riemann mapping theorem（smooth boundary）379

5 The Dirichlet problem on a domain with a rough boundary383

6 The Riemann mapping theorem（rough boundary）398

7 The Neumann boundary problem402

8 The Hodge decomposition and harmonic forms410

9 Natural boundary problems for the Hodge Laplacian421

10 Isothermal coordinates and conformal structures on surfaces438

11 General elliptic boundary problems441

12 Operator properties of regular boundary problems462

A Spaces of generalized functions on manifolds with boundary471

B The Mayer-Vietoris sequence in de Rham cohomology475

References478

6 Linear Evolution Equations481

1 The heat equation and the wave equation on bounded domains482

2 The heat equation and wave equation on unbounded domains490

3 Maxwell's equations496

4 The Cauchy-Kowalewsky theorem499

5 Hyperbolic systems504

6 Geometrical optics510

7 The formation of caustics518

8 Boundary layer phenomena for the heat semigroup535

A Some Banach spaces of harmonic functions541

B The stationary phase method543

References545

A Outline of Functional Analysis549

1 Banach spaces549

2 Hilbert spaces556

3 Fréchet spaces;locally convex spaces561

4 Duality564

5 Linear operators571

6 Compact operators579

7 Fredholm operators593

8 Unbounded operators596

9 Semigroups603

References615

B Manifolds,Vector Bundles,and Lie Groups617

1 Metric spaces and topological spaces617

2 Manifolds622

3 Vector bundles624

4 Sard's theorem626

5 Lie groups627

6 The Campbell-Hausdorff formula630

7 Representations of Lie groups and Lie algebras632

8 Representations of compact Lie groups636

9 Representations of SU（2）and related groups641

References647

Index649