The Theory of Partial Differential Equations

Front Cover
CUP Archive, Aug 2, 1973 - Mathematics - 490 pages
The basis of this graduate-level textbook is a careful survey of a wide range of problems affecting the solution of linear partial differential equations. The book begins with a fairly elementary introduction to the theory of Fourier series of continuous functions and goes on to describe the fundamental theory of linear partial differential equations of elliptic and hyperbolic types, equations of evolution, semi-linear hyperbolic equations and selected topics on Green's functions and spectra of some special operators. The book is intended for use by pure mathematicians in functional analysis. The selection of material is interesting and differs from existing literature in European languages. This paperback edition will make it particularly attractive to graduate students in pure and applied mathematics.
 

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Contents

N Chapter 1 Fourier series and Fourier transforms 1 Fourier series p
1
Dirichlets integrals p
4
Application to the heat transfer equation p
12
The system of orthogonal functions in L 12 p
15
Fouriers integral formula p
20
Fourier transforms p
23
Several variables and functional spaces p
27
Multiple Fourier series p
31
The existence theorem of solutions p
272
Phenomena with a finite speed of propagation p
276
Solution of wave equations p
278
Systems of hyperbolic firstorder equations p
282
Evolution equations 1 Introduction p
286
Laplace transforms and semigroups p
291
Parabolic semigroups p
300
Semigroups for selfadjoint operators p
306

Fourier transform of several variables p
38
Plancherels theorem p
41
Plancherels theorem reconsidered p
44
Distributions 1 Definition of distribution and convergence of sequence of distributions p
47
Fundamental properties of Fréchet spaces p
58
Function spaces LOM92 OLOM2 p
70
Structures of Olom92 and p
83
Fourier transforms of distributions p
96
Concrete examples of Fourier transforms p
107
The relationship of Fourier transforms and convolutions p
117
Laplace transforms of functions p
123
Laplace transforms of distributions p
127
Laplace transforms of vectorvalued functions p
135
Fourier transform of a spherically symmetric function p
138
Elementary solutions for elliptic operators with constant coefficients p
142
Elliptic equations fundamental theory 1 Introduction p
147
The solution of the Dirichlet problem Greens operator p
150
The theorem of Rellich p
172
Trace on boundaries boundary values in the wider sense p
176
Characterization of DL2+2 p
187
Properties of Lam12 p
189
Improving the estimation of yf p
190
Boundary value problems for elliptic differential equations of second order p
195
Dirichlet problems for the general secondorder elliptic operators p
199
Fredholms alternative theorem for a completely continuous operator p
206
Differentiability of a solution p
210
Differentiability of a solution in the neighbourhood of a boundary p
217
The interpolation theorem of LamR+ p
227
Some remarks on Dirichlet problems p
231
The boundary value problem of the third kind p
234
Extension of selfadjoint operators p
237
The Dirichlet problem for an elliptic operator of higher order p
240
Initial value problems Cauchy problems 1 Introduction p
243
The CauchyKowalewski and Holmgren theorems p
245
Notes on the solubility of the Cauchy problem p
252
Local solubility of a Cauchy problem p
256
The continuity of solutions for initial value problem p
261
Dependence domain p
269
Two examples of parabolic equations p
308
Hyperbolic equations 1 Introduction p
311
Remarks on energy inequalities p
316
Existence theorem 1 for a solution of a system of symmetric hyperbolic equa tions the case in which the coefficients are independent of t p
320
Existence theorem 2 for a system of symmetric hyperbolic equations general case p
329
Nonsymmetric hyperbolic systems p
336
Singular integral operator p
338
Properties of singular integral operators p
343
Energy inequality for a system of hyperbolic equations p
348
Energy inequality for hyperbolic equations p
355
The existence theorem for the solution of a system of hyperbolic equations p
357
Dependence domain p
361
Existence theorem for the solution of a hyperbolic equation p
366
Uniqueness of the solution of a Cauchy problem p
368
Semilinear hyperbolic equations 1 Introduction p
377
Smoothness of a composed function p
383
Existence theorem 1 the case of the hyperbolic system p
386
Existence theorem 2 case of a single equation p
392
Example semilinear wave equation p
395
Greens functions and spectra 1 Introduction p
400
Greens function for 41 p
405
Fredholms theorem p
406
Concrete construction of a Greens function p
413
Properties of Greens functions p
424
The solution of a wave equation in the exterior domain p
430
Discrete spectrum for Schrödingers operator p
436
Friedrichs extension p
438
Discrete spectrum p
444
The finiteness of a spectrum in the negative part p
447
Selfadjoint extensions p
450
Negative spectrum of 4+cx p
453
Supplementary remarks 1 General boundary value problems of highorder elliptic equations p
457
Completeness of a system of eigenfunctions p
465
Guide to the literature
471
Bibliography
478
Symbols
484
Index
487
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