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Partial Differential Equations II : Elements of the Modern Theory. Equations with Constant Coefficients - Yu.V. Egorov

Partial Differential Equations II

Elements of the Modern Theory. Equations with Constant Coefficients

Paperback

Published: 20th May 1999
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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

"

Prefacep. 4
Notationp. 5
Pseudodifferential Operatorsp. 6
Definition and Simplest Propertiesp. 6
The Expression for an Operator in Terms of Amplitude. The Connection Between the Amplitude and the Symbol. Symbols of Transpose and Adjoint Operatorsp. 9
The Composition Theorem. The Parametrix of an Elliptic Operatorp. 14
Action of Pseudodifferential Operators in Sobolev Spaces and Precise Regularity Theorems for Solutions of Elliptic Equationsp. 17
Change of Variables and Pseudodifferential Operators on a Manifoldp. 19
Formulation of the Index Problem. The Simplest Index Formulaep. 24
Ellipticity with a Parameter. Resolvent and Complex Powers of Elliptic Operatorsp. 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 Boundaryp. 36
Definition and Boundedness Theoremsp. 36
Smoothness of Solutions of Second-order Elliptic Equationsp. 37
Connection with Pseudodifferential Operatorsp. 37
DiagonaLization of Hyperbolic System of Equationsp. 38
Calderon's Theoremp. 39
Reduction of the Oblique Derivative Problem to a Problem on the Boundaryp. 40
Reduction of the Boundary-value Problem for the Second-order Equation to a Problem on the Boundaryp. 41
Reduction of the Boundary-value Problem for an Elliptic System to a Problem on the Boundaryp. 43
Wave Front of a Distribution and Simplest Theorems on Propagation of Singularitiesp. 44
Definition and Examplesp. 44
Properties of the Wave Front Setp. 45
Applications to Differential Equationsp. 47
Some Generalizationsp. 48
Fourier Integral Operatorsp. 48
Definition and Examplesp. 48
Some Properties of Fourier Integral Operatorsp. 50
Composition of Fourier Integral Operators with Pseudodifferential Operatorsp. 52
Canonical Transformationsp. 53
Connection Between Canonical Transformations and Fourier Integral Operatorsp. 55
Lagrangian Manifolds and Phase Functionsp. 57
Lagrangian Manifolds and Fourier Distributionsp. 59
Global Definition of a Fourier Integral Operatorp. 59
Pseudodifferential Operators of Principal Typep. 60
Definition and Examplesp. 60
Operators with Real Principal Symbolp. 61
Solvability of Equations of Principle Type with Real Principal Symbolp. 63
Solvability of Operators of Principal Type with Complex-valued Principal Symbolp. 64
Mixed Problems for Hyperbolic Equationsp. 65
Formulation of the Problemp. 65
The Hersh-Kreiss Conditionp. 66
The Sakamoto Conditionsp. 68
Reflection of Singularities on the Boundaryp. 69
Friedlander's Examplep. 71
Application of Canonical Transformationsp. 73
Classification of Boundary Pointsp. 74
Taylor's Examplep. 74
Oblique Derivative Problemp. 75
Method of Stationary Phase and Short-wave Asymptoticsp. 78
Method of Stationary Phasep. 79
Local Asymptotic Solutions of Hyperbolic Equationsp. 82
Cauchy Problem with Rapidly Oscillating Initial Datap. 86
Local Parametrix of the Cauchy Problem and Propagation of Singularities of Solutionsp. 87
The Maslov Canonical Operator and Global Asymptotic Solutions of the Cauchy Problemp. 90
Asymptotics of Eigenvalues of Self-adjoint Differential and Pseudodifferential Operatorsp. 96
Variational Principles and Estimates for Eigenvaluesp. 96
Asymptotics of the Eigenvalues of the Laplace Operator in a Euclidean Domainp. 99
General Formula of Weyl Asymptotics and the Method of Approximate Spectral Projectionp. 102
Tauberian Methodsp. 106
The Hyperbolic Equation Methodp. 110
Bibliographical Commentsp. 113
Referencesp. 114
Prefacep. 125
Generalized Functions and Fundamental Solutions of Differential Equationsp. 128
Generalized Functions and Operations on themp. 128
Differentiation of Generalized Functionsp. 128
Change of Variables in Generalized Functionsp. 130
Support of a Generalized Functionp. 134
Singular Support of Generalized Functionsp. 136
The Convolution of Generalized Functionsp. 136
Boundary Values of Analytic Functionsp. 139
The Space of Tempered Distributionsp. 141
Fundamental Solutions of Differential Equationsp. 142
The Fundamental Solutionsp. 142
Examples of Fundamental Solutionsp. 143
The Propagation of Wavesp. 146
The Construction of Fundamental Solutions of Ordinary Differential Equationsp. 147
A Mean Value Theoremp. 148
Fourier Transformation of Generalized Functionsp. 149
Fourier Transformation of Test Functionsp. 149
Fourier Transformation of Rapidly Decreasing Functionsp. 149
Properties of the Fourier Transformationp. 149
Fourier Transformation of Functions with Compact Supportp. 150
Fourier Transformation of Tempered Generalized Functionsp. 151
Closure of the Fourier Transformation with Respect to Continuityp. 151
Propertiesof the Fourier Transformationp. 151
Methods for Computing Fourier Transformsp. 153
Examples of the Computation of Fourier Transformsp. 154
The Sobolev Function Spacesp. 155
Fourier Transformation of Rapidly Growing Generalized Functionsp. 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 Transformationp. 158
Analytic Functionalsp. 158
The Paley-Wiener Theoryp. 160
Fourier Transform of Generalized Functions with Compact Supportsp. 160
Tempered Distributions with Support in a Conep. 160
Exponentially Growing Distributions Having Support in a Conep. 161
Convolution and Fourier Transformp. 163
Existence and Uniqueness of Solutions of Differential Equationsp. 164
The Problem of Divisionp. 164
The Problem of Division in Classes of Rapidly Growing Distributionsp. 164
The Problem of Division in Classes of Exponentially Growing Generalized Functions. The Hörmander Staircasep. 166
The Problem of Division in Classes of Tempered Distributionsp. 167
Regularization. The Methods of "Subtraction" and Exit to the Complex Domain and the Riesz Power Methodp. 168
The Method of Subtractionp. 169
The Method of Exit to the Complex Domainp. 171
The Riesz Method of Complex Powersp. 172
Equations in a Convex Cone. An Operational Calculusp. 173
Equations in a Conep. 173
An Operational Calculusp. 175
Differential-difference Equations on a Semi-axisp. 177
Propagation of Singularities and Smoothness of Solutionsp. 178
Characteristics of Differential Equationsp. 178
Wave Fronts Bicharacteristics and Propagation of Singularitiesp. 180
Smoothness of Solutions of Elliptic Equations. Hypoellipticityp. 183
Smoothness of Generalized Solutions of Elliptic Equationsp. 183
Hypoelliptic Operatorsp. 184
The Function <$>P_+^\lambda<$> for Polynomials of Second-degree and its Application in the Construction of Fundamental Solutionsp. 186
The Function <$>P_+^\lambda<$> for the Case when P is a Real Linear Functionp. 186
Analytic Continuation with Respect to ¿p. 186
An Application to Bessel Functionsp. 188
The Function <$>P_+^\lambda<$> for the Case when P(x) is a Quadratic Form of the Type (m, n - m) with Real Coefficientsp. 188
The Case m = np. 189
Application to Decomposition of ¿-Function into Plane Wavesp. 190
The Case <$>1 \leqslant m \leqslant n - 1<$>p. 191
Application to Bessel Functionsp. 193
Invariant Fundamental Solutions of Second-order Equations with Real Coefficientsp. 196
Analysis of Invariance Properties of the Equationp. 197
Determination of the Regular Part of an Invariant Fundamental Solutionp. 198
Regularization of the Formal Fundamental Solution for the Case q = 0p. 200
The Case m = 0 or m = np. 200
The Case <$>1 \leqslant m \leqslant n - 1<$>p. 201
Regularization of the Fundamental Solution for the Case q &neq; 0p. 204
The Case <$>1 \leqslant m \leqslant n - 1<$>p. 204
The Case m = 0 or m = np. 207
On Singularities of Fundamental Solutions of Second-order Equations with Real Coefficients and with Non-degenerate Quadratic Formp. 211
Boundary-value Problems in Half-spacep. 212
Equations with Constant Coefficients in a Half-spacep. 213
General Solution of Equation (0.1) in a Half-spacep. 213
Classification of Equations in Half-spacep. 215
Regular Boundary-value Problems in a Half-space in Classes of Bounded Functionsp. 220
Regular Boundary-value Problemsp. 221
Examples of Regular Boundary-value Problemsp. 224
Regular Boundary-value Problems in Classes of Exponentially Growing Functionsp. 226
Definition and Examplesp. 226
The Cauchy Problemp. 228
The Dirichlet Problem for Elliptic Equationsp. 229
Regular Boundary-value Problems in the Class of Functions of Arbitrary Growthp. 229
Well-posed and Continuous Boundary-value Problems in a Half-spacep. 231
Well-posed Boundary Value Problemsp. 231
Continuous Well-posed Boundary-value Problemsp. 232
The Poisson Kernel for the Boundary-value Problem in a Half-spacep. 234
The Poisson Kernel and the Fundamental Solution of the Boundary-value Problemp. 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 Equationsp. 238
Non-homogeneous Equations in a Half-spacep. 238
Boundary-value Problems for Non-homogeneous Equationsp. 240
Sharp and Diffusion Fronts of Hyperbolic Equationsp. 240
Basic Notionsp. 241
The Petrovskij Criterionp. 244
The Local Petrovskij Criterionp. 246
Geometry of Lacunae Near Concrete Singularities of Frontsp. 247
Equations with Variable Coefficientsp. 250
Bibliographical Commentsp. 250
Referencesp. 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