## Partial Differential Equations: Basic TheoryMathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modem as weIl as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sci ences (AMS) series, which will focus on advanced textbooks and research level monographs. |

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### Contents

Basic Theory of ODE and Vector Fields | 1 |

1 The derivative | 2 |

2 Fundamental local existence theorem for ODE | 8 |

3 Inverse function and implicit function theorems | 11 |

4 Constantcoefficient linear systems exponentiation of matrices | 14 |

Duhamels principle | 24 |

6 Dependence of solutions on initial data and on other parameters | 28 |

7 Flows and vector fields | 31 |

4 Sobolev spaces on bounded domains | 284 |

5 The Sobolev spaces | 291 |

6 The Schwartz kernel theorem | 296 |

References | 300 |

Linear Elliptic Equations | 302 |

1 Existence and regularity of solutions to the Dirichlet problem | 303 |

2 The weak and strong maximum principles | 311 |

3 The Dirichlet problem on the ball in R | 319 |

8 Lie brackets | 36 |

9 Commuting flows Frobeniuss theorem | 39 |

10 Hamiltonian systems | 43 |

11 Geodesies | 45 |

12 Variational problems and the stationary action principle | 53 |

13 Differential forms | 62 |

14 The symplectic form and canonical transformations | 73 |

15 Firstorder scalar nonlinear PDE | 79 |

16 Completely integrable Hamiltonian systems | 85 |

17 Examples of integrable systems central force problems | 90 |

18 Relativistic motion | 93 |

19 Topological applications of differential forms | 97 |

20 Critical points and index of a vector field | 105 |

A Nonsmooth vector fields | 109 |

References | 112 |

The Laplace Equation and Wave Equation | 114 |

1 Vibrating strings and membranes | 116 |

2 The divergence of a vector field | 125 |

3 The covariant derivative and divergence of tensor fields | 130 |

4 The Laplace operator on a Riemannian manifold | 137 |

5 The wave equation on a product manifold and energy conservation | 140 |

6 Uniqueness and finite propagation speed | 145 |

7 Lorentz manifolds and stressenergy tensors | 149 |

8 More general hyperbolic equations energy estimates | 154 |

9 The symbol of a differential operator and a general GreenStokes formula | 158 |

10 The Hodge Laplacian on kforms | 161 |

11 Maxwells equations | 165 |

References | 173 |

Fourier Analysis Distributions and ConstantCoefficient Linear PDE | 176 |

1 Fourier series | 177 |

2 Harmonic functions and holomorphic functions in the plane | 187 |

3 The Fourier transform | 197 |

4 Distributions and tempered distributions | 204 |

5 The classical evolution equations | 216 |

6 Radial distributions polar coordinates and Bessel functions | 225 |

7 The method of images and Poissons summation formula | 234 |

8 Homogeneous distributions and principal value distributions | 239 |

9 Elliptic operators | 245 |

10 Local solvability of constantcoefficient PDE | 248 |

11 The discrete Fourier transform | 250 |

12 The fast Fourier transform | 258 |

The mighty Gaussian and the sublime gamma function | 262 |

References | 268 |

Sobolev Spaces | 270 |

2 The complex interpolation method | 276 |

3 Sobolev spaces on compact manifolds | 281 |

4 The Riemann mapping theorem smooth boundary | 324 |

5 The Dirichlet problem on a domain with a rough boundary | 327 |

6 The Riemann mapping theorem rough boundary | 341 |

7 The Neumann boundary problem | 345 |

8 The Hodge decomposition and harmonic forms | 352 |

9 Natural boundary problems for the Hodge Laplacian | 362 |

10 Isothermal coordinates and conformal structures on surfaces | 377 |

11 General elliptic boundary problems | 380 |

12 Operator properties of regular boundary problems | 398 |

Spaces of generalized functions on manifolds with boundary | 406 |

The Mayer Vietoris sequence in deRham cohomology | 409 |

References | 412 |

Linear Evolution Equations | 415 |

1 The heat equation and the wave equation on bounded domains | 416 |

2 The heat equation and wave equation on unbounded domains | 423 |

3 Maxwells equations | 428 |

4 The CauchyKowalewsky theorem | 431 |

5 Hyperbolic systems | 435 |

6 Geometrical optics | 441 |

7 The formation of caustics | 448 |

Some Banach spaces of harmonic functions | 464 |

The stationary phase method | 465 |

References | 467 |

Outline of Functional Analysis | 469 |

2 Hilbert spaces | 476 |

3 Frechet spaces locally convex spaces | 480 |

4 Duality | 482 |

5 Linear operators | 487 |

6 Compact operators | 495 |

7 Fredholm operators | 507 |

8 Unbounded operators | 511 |

9 Semigroups | 516 |

References | 527 |

Manifolds Vector Bundles and Lie Groups Introduction | 529 |

1 Metric spaces and topological spaces | 530 |

2 Manifolds | 535 |

3 Vector bundles | 536 |

4 Sards theorem | 538 |

5 Lie groups | 539 |

6 The CampbellHausdorff formula | 542 |

7 Representations of Lie groups and Lie algebras | 544 |

8 Representations of compact Lie groups | 547 |

9 Representations of SU2 and related groups | 551 |

557 | |

Index | 559 |

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### Common terms and phrases

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