## The Theory of Partial Differential EquationsThe 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 |

478 | |

Symbols | 484 |

487 | |

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Page 478 - The fundamental solution of linear elliptic differential equations with analytic coefficients, Comm.