This book, the first printing of which was published as Volume 31 of the Encyclopaedia of Mathematical Sciences, contains a survey of the modern theory of general linear partial differential equations and a detailed review of equations with constant coefficients. Readers will be interested in an introduction to microlocal analysis and its applications including singular integral operators, pseudodifferential operators, Fourier integral operators and wavefronts, a survey of the most important results about the mixed problem for hyperbolic equations, a review of asymptotic methods including short wave asymptotics, the Maslov canonical operator and spectral asymptotics, a detailed description of the applications of distribution theory to partial differential equations with constant coefficients including numerous interesting special topics.
..". Summarizing, the book presents a well-written survey of the modern theory of general PDE-s with most necessary proofs or hints, and with some explicit constructions. Therefore it will be useful for specialists in PDE-s as well as for students with basic knowledge in classical theory of PDE-s. The book is warmly recommended also to physicists, engineers and anyone interested in theory or/and applications of PDE-s."
J. Hegedus, Acta Scientiarum Mathematicarum, Vol. 70, 2004
Preface | p. 4 |
Notation | p. 5 |
Pseudodifferential Operators | p. 6 |
Definition and Simplest Properties | p. 6 |
The Expression for an Operator in Terms of Amplitude. The Connection Between the Amplitude and the Symbol. Symbols of Transpose and Adjoint Operators | p. 9 |
The Composition Theorem. The Parametrix of an Elliptic Operator | p. 14 |
Action of Pseudodifferential Operators in Sobolev Spaces and Precise Regularity Theorems for Solutions of Elliptic Equations | p. 17 |
Change of Variables and Pseudodifferential Operators on a Manifold | p. 19 |
Formulation of the Index Problem. The Simplest Index Formulae | p. 24 |
Ellipticity with a Parameter. Resolvent and Complex Powers of Elliptic Operators | p. 26 |
Pseudodifferential Operators in <$>{\op R}^n<$> | p. 32 |
Singular Integral Operators and their Applications. Calderon's Theorem. Reduction of Boundary-value Problems for Elliptic Equations to Problems on the Boundary | p. 36 |
Definition and Boundedness Theorems | p. 36 |
Smoothness of Solutions of Second-order Elliptic Equations | p. 37 |
Connection with Pseudodifferential Operators | p. 37 |
DiagonaLization of Hyperbolic System of Equations | p. 38 |
Calderon's Theorem | p. 39 |
Reduction of the Oblique Derivative Problem to a Problem on the Boundary | p. 40 |
Reduction of the Boundary-value Problem for the Second-order Equation to a Problem on the Boundary | p. 41 |
Reduction of the Boundary-value Problem for an Elliptic System to a Problem on the Boundary | p. 43 |
Wave Front of a Distribution and Simplest Theorems on Propagation of Singularities | p. 44 |
Definition and Examples | p. 44 |
Properties of the Wave Front Set | p. 45 |
Applications to Differential Equations | p. 47 |
Some Generalizations | p. 48 |
Fourier Integral Operators | p. 48 |
Definition and Examples | p. 48 |
Some Properties of Fourier Integral Operators | p. 50 |
Composition of Fourier Integral Operators with Pseudodifferential Operators | p. 52 |
Canonical Transformations | p. 53 |
Connection Between Canonical Transformations and Fourier Integral Operators | p. 55 |
Lagrangian Manifolds and Phase Functions | p. 57 |
Lagrangian Manifolds and Fourier Distributions | p. 59 |
Global Definition of a Fourier Integral Operator | p. 59 |
Pseudodifferential Operators of Principal Type | p. 60 |
Definition and Examples | p. 60 |
Operators with Real Principal Symbol | p. 61 |
Solvability of Equations of Principle Type with Real Principal Symbol | p. 63 |
Solvability of Operators of Principal Type with Complex-valued Principal Symbol | p. 64 |
Mixed Problems for Hyperbolic Equations | p. 65 |
Formulation of the Problem | p. 65 |
The Hersh-Kreiss Condition | p. 66 |
The Sakamoto Conditions | p. 68 |
Reflection of Singularities on the Boundary | p. 69 |
Friedlander's Example | p. 71 |
Application of Canonical Transformations | p. 73 |
Classification of Boundary Points | p. 74 |
Taylor's Example | p. 74 |
Oblique Derivative Problem | p. 75 |
Method of Stationary Phase and Short-wave Asymptotics | p. 78 |
Method of Stationary Phase | p. 79 |
Local Asymptotic Solutions of Hyperbolic Equations | p. 82 |
Cauchy Problem with Rapidly Oscillating Initial Data | p. 86 |
Local Parametrix of the Cauchy Problem and Propagation of Singularities of Solutions | p. 87 |
The Maslov Canonical Operator and Global Asymptotic Solutions of the Cauchy Problem | p. 90 |
Asymptotics of Eigenvalues of Self-adjoint Differential and Pseudodifferential Operators | p. 96 |
Variational Principles and Estimates for Eigenvalues | p. 96 |
Asymptotics of the Eigenvalues of the Laplace Operator in a Euclidean Domain | p. 99 |
General Formula of Weyl Asymptotics and the Method of Approximate Spectral Projection | p. 102 |
Tauberian Methods | p. 106 |
The Hyperbolic Equation Method | p. 110 |
Bibliographical Comments | p. 113 |
References | p. 114 |
Preface | p. 125 |
Generalized Functions and Fundamental Solutions of Differential Equations | p. 128 |
Generalized Functions and Operations on them | p. 128 |
Differentiation of Generalized Functions | p. 128 |
Change of Variables in Generalized Functions | p. 130 |
Support of a Generalized Function | p. 134 |
Singular Support of Generalized Functions | p. 136 |
The Convolution of Generalized Functions | p. 136 |
Boundary Values of Analytic Functions | p. 139 |
The Space of Tempered Distributions | p. 141 |
Fundamental Solutions of Differential Equations | p. 142 |
The Fundamental Solutions | p. 142 |
Examples of Fundamental Solutions | p. 143 |
The Propagation of Waves | p. 146 |
The Construction of Fundamental Solutions of Ordinary Differential Equations | p. 147 |
A Mean Value Theorem | p. 148 |
Fourier Transformation of Generalized Functions | p. 149 |
Fourier Transformation of Test Functions | p. 149 |
Fourier Transformation of Rapidly Decreasing Functions | p. 149 |
Properties of the Fourier Transformation | p. 149 |
Fourier Transformation of Functions with Compact Support | p. 150 |
Fourier Transformation of Tempered Generalized Functions | p. 151 |
Closure of the Fourier Transformation with Respect to Continuity | p. 151 |
Propertiesof the Fourier Transformation | p. 151 |
Methods for Computing Fourier Transforms | p. 153 |
Examples of the Computation of Fourier Transforms | p. 154 |
The Sobolev Function Spaces | p. 155 |
Fourier Transformation of Rapidly Growing Generalized Functions | p. 156 |
Functions on the Space <$>Z({\op C}^n)<$> | p. 156 |
Fourier Transformation on the Space <$>{\cal D}^\prime ({\op R}^n)<$> | p. 157 |
Operations on the Space <$>Z^\prime ({\op C}^n)<$> | p. 158 |
Properties of the Fourier Transformation | p. 158 |
Analytic Functionals | p. 158 |
The Paley-Wiener Theory | p. 160 |
Fourier Transform of Generalized Functions with Compact Supports | p. 160 |
Tempered Distributions with Support in a Cone | p. 160 |
Exponentially Growing Distributions Having Support in a Cone | p. 161 |
Convolution and Fourier Transform | p. 163 |
Existence and Uniqueness of Solutions of Differential Equations | p. 164 |
The Problem of Division | p. 164 |
The Problem of Division in Classes of Rapidly Growing Distributions | p. 164 |
The Problem of Division in Classes of Exponentially Growing Generalized Functions. The HÃ¶rmander Staircase | p. 166 |
The Problem of Division in Classes of Tempered Distributions | p. 167 |
Regularization. The Methods of "Subtraction" and Exit to the Complex Domain and the Riesz Power Method | p. 168 |
The Method of Subtraction | p. 169 |
The Method of Exit to the Complex Domain | p. 171 |
The Riesz Method of Complex Powers | p. 172 |
Equations in a Convex Cone. An Operational Calculus | p. 173 |
Equations in a Cone | p. 173 |
An Operational Calculus | p. 175 |
Differential-difference Equations on a Semi-axis | p. 177 |
Propagation of Singularities and Smoothness of Solutions | p. 178 |
Characteristics of Differential Equations | p. 178 |
Wave Fronts Bicharacteristics and Propagation of Singularities | p. 180 |
Smoothness of Solutions of Elliptic Equations. Hypoellipticity | p. 183 |
Smoothness of Generalized Solutions of Elliptic Equations | p. 183 |
Hypoelliptic Operators | p. 184 |
The Function <$>P_+^\lambda<$> for Polynomials of Second-degree and its Application in the Construction of Fundamental Solutions | p. 186 |
The Function <$>P_+^\lambda<$> for the Case when P is a Real Linear Function | p. 186 |
Analytic Continuation with Respect to Â¿ | p. 186 |
An Application to Bessel Functions | p. 188 |
The Function <$>P_+^\lambda<$> for the Case when P(x) is a Quadratic Form of the Type (m, n - m) with Real Coefficients | p. 188 |
The Case m = n | p. 189 |
Application to Decomposition of Â¿-Function into Plane Waves | p. 190 |
The Case <$>1 \leqslant m \leqslant n - 1<$> | p. 191 |
Application to Bessel Functions | p. 193 |
Invariant Fundamental Solutions of Second-order Equations with Real Coefficients | p. 196 |
Analysis of Invariance Properties of the Equation | p. 197 |
Determination of the Regular Part of an Invariant Fundamental Solution | p. 198 |
Regularization of the Formal Fundamental Solution for the Case q = 0 | p. 200 |
The Case m = 0 or m = n | p. 200 |
The Case <$>1 \leqslant m \leqslant n - 1<$> | p. 201 |
Regularization of the Fundamental Solution for the Case q &neq; 0 | p. 204 |
The Case <$>1 \leqslant m \leqslant n - 1<$> | p. 204 |
The Case m = 0 or m = n | p. 207 |
On Singularities of Fundamental Solutions of Second-order Equations with Real Coefficients and with Non-degenerate Quadratic Form | p. 211 |
Boundary-value Problems in Half-space | p. 212 |
Equations with Constant Coefficients in a Half-space | p. 213 |
General Solution of Equation (0.1) in a Half-space | p. 213 |
Classification of Equations in Half-space | p. 215 |
Regular Boundary-value Problems in a Half-space in Classes of Bounded Functions | p. 220 |
Regular Boundary-value Problems | p. 221 |
Examples of Regular Boundary-value Problems | p. 224 |
Regular Boundary-value Problems in Classes of Exponentially Growing Functions | p. 226 |
Definition and Examples | p. 226 |
The Cauchy Problem | p. 228 |
The Dirichlet Problem for Elliptic Equations | p. 229 |
Regular Boundary-value Problems in the Class of Functions of Arbitrary Growth | p. 229 |
Well-posed and Continuous Boundary-value Problems in a Half-space | p. 231 |
Well-posed Boundary Value Problems | p. 231 |
Continuous Well-posed Boundary-value Problems | p. 232 |
The Poisson Kernel for the Boundary-value Problem in a Half-space | p. 234 |
The Poisson Kernel and the Fundamental Solution of the Boundary-value Problem | p. 234 |
The Connection Between the Fundamental Solution of the Cauchy Problem and the Retarded Fundamental Solution of the Operator P(∂_{x}) | p. 235 |
Boundary-value Problems in a Half-space for Non-homogeneous Equations | p. 238 |
Non-homogeneous Equations in a Half-space | p. 238 |
Boundary-value Problems for Non-homogeneous Equations | p. 240 |
Sharp and Diffusion Fronts of Hyperbolic Equations | p. 240 |
Basic Notions | p. 241 |
The Petrovskij Criterion | p. 244 |
The Local Petrovskij Criterion | p. 246 |
Geometry of Lacunae Near Concrete Singularities of Fronts | p. 247 |
Equations with Variable Coefficients | p. 250 |
Bibliographical Comments | p. 250 |
References | p. 251 |
Table of Contents provided by Publisher. All Rights Reserved. |
ISBN: 9783540653776
ISBN-10: 3540653775
Audience:
General
Format:
Paperback
Language:
English
Number Of Pages: 266
Published: 20th May 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.52 x 15.6
x 1.6
Weight (kg): 0.41