Partial Differential Equations: Basic Theory

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Springer Science & Business Media, Jun 25, 1996 - Mathematics - 563 pages
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Mathematics 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
References
557
Index
559
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About the author (1996)

Taylor, Reader in Government, University of Essex.