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# Partial Differential Equations II

### 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

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 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 Hörmander 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 ¿ 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 ¿-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. 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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