ISBN: 3-540-65377-5
TITLE: Elements of the Modern Theory of Partial Differential Equations
AUTHOR: Egorov, Yu.V.; Komech, A.I.; Shubin, M.A.
TOC:

I. Linear Partial Differential Equations. Elements of the Modern Theory
Yu.V. Egorov, M.A. Shubin
Translated from the Russian by P.C. Sinha
Preface 4
Notation 5
 1. Pseudodifferential Operators 6
1.1. Definition and Simplest Properties 6
1.2. The Expression for an Operator in Terms of Amplitude. The Connection Between the Amplitude and the Symbol. Symbols of Transpose and Adjoint Operators 9
1.3. The Composition Theorem. The Parametrix of an Elliptic Operator 14
1.4. Action of Pseudodifferential Operators in Sobolev Spaces and Precise Regularity Theorems for Solutions of Elliptic Equations 17
1.5. Change of Variables and Pseudodifferential Operators on a Manifold 19
1.6. Formulation of the Index Problem. The Simplest Index Formulae 24
1.7. Ellipticity with a Parameter. Resolvent and Complex Powers of Elliptic Operators 26
1.8. Pseudodifferential Operators in R^n 32
 2. Singular Integral Operators and their Applications. Calderon's Theorem. Reduction of Boundary-value Problems for Elliptic Equations to Problems on the Boundary 36
2.1. Definition and Boundedness Theorems 36
2.2. Smoothness of Solutions of Second-order Elliptic Equations 37
2.3. Connection with Pseudodifferential Operators 37
2.4. DiagonaLization of Hyperbolic System of Equations 38
2.5. Calderon's Theorem 39
2.6. Reduction of the Oblique Derivative Problem to a Problem on the Boundary 40
2.7. Reduction of the Boundary-value Problem for the Second-order Equation to a Problem on the Boundary 41
2.8. Reduction of the Boundary-value Problem for an Elliptic System to a Problem on the Boundary 43
 3. Wave Front of a Distribution and Simplest Theorems on Propagation of Singularities 44
3.1. Definition and Examples 44
3.2. Properties of the Wave Front Set 45
3.3. Applications to Differential Equations 47
3.4. Some Generalizations 48
 4. Fourier Integral Operators 48
4.1. Definition and Examples 48
4.2. Some Properties of Fourier Integral Operators 50
4.3. Composition of Fourier Integral Operators with Pseudodifferential Operators 52
4.4. Canonical Transformations 53
4.5. Connection Between Canonical Transformations and Fourier Integral Operators 55
4.6. Lagrangian Manifolds and Phase Functions 57
4.7. Lagrangian Manifolds and Fourier Distributions 59
4.8. Global Definition of a Fourier Integral Operator 59
 5. Pseudodifferential Operators of Principal Type 60
5.1. Definition and Examples 60
5.2. Operators with Real Principal Symbol 61
5.3. Solvability of Equations of Principle Type with Real Principal Symbol 63
5.4. Solvability of Operators of Principal Type with Complex-valued Principal Symbol 64
 6. Mixed Problems for Hyperbolic Equations 65
6.1. Formulation of the Problem 65
6.2. The Hersh-Kreiss Condition 66
6.3. The Sakamoto Conditions 68
6.4. Reflection of Singularities on the Boundary 69
6.5. Friedlander's Example 71
6.6. Application of Canonical Transformations 73
6.7. Classification of Boundary Points 74
6.8. Taylor's Example 74
6.9. Oblique Derivative Problem 75
 7. Method of Stationary Phase and Short-wave Asymptotics 78
7.1. Method of Stationary Phase 79
7.2. Local Asymptotic Solutions of Hyperbolic Equations 82
7.3. Cauchy Problem with Rapidly Oscillating Initial Data 86
7.4. Local Parametrix of the Cauchy Problem and Propagation of Singularities of Solutions 87
7.5. The Maslov Canonical Operator and Global Asymptotic Solutions of the Cauchy Problem 90
 8. Asymptotics of Eigenvalues of Self-adjoint Differential and Pseudodifferential Operators 96
8.1. Variational Principles and Estimates for Eigenvalues 96
8.2. Asymptotics of the Eigenvalues of the Laplace Operator in a Euclidean Domain 99
8.3. General Formula of Weyl Asymptotics and the Method of Approximate Spectral Projection 102
8.4. Tauberian Methods 106
8.5. The Hyperbolic Equation Method 110
Bibliographical Comments 113
References 114
II. Linear Partial Differential Equations with Constant Coefficients
A.I. Komech
Translated from the Russian by P.C. Sinha
Preface 125
Chapter 1. Generalized Functions and Fundamental Solutions
of Differential Equations 128
 1. Generalized Functions and Operations on them 128
1.1. Differentiation of Generalized Functions 128
1.2. Change of Variables in Generalized Functions 130
1.3. Support of a Generalized Function 134
1.4. Singular Support of Generalized Functions 136
1.5. The Convolution of Generalized Functions 136
1.6. Boundary Values of Analytic Functions 139
1.7. The Space of Tempered Distributions 141
 2. Fundamental Solutions of Differential Equations 142
2.1. The Fundamental Solutions 142
2.2. Examples of Fundamental Solutions 143
2.3. The Propagation of Waves 146
2.4. The Construction of Fundamental Solutions of Ordinary Differential Equations 147
2.5. A Mean Value Theorem 148
Chapter 2. Fourier Transformation of Generalized Functions 149
 1. Fourier Transformation of Test Functions 149
1.1. Fourier Transformation of Rapidly Decreasing Functions 149
1.2. Properties of the Fourier Transformation 149
1.3. Fourier Transformation of Functions with Compact Support 150
 2. Fourier Transformation of Tempered Generalized Functions 151
2.1. Closure of the Fourier Transformation with Respect to Continuity 151
2.2. Properties of the Fourier Transformation 151
2.3. Methods for Computing Fourier Transforms 153
2.4. Examples of the Computation of Fourier Transforms 154
 3. The Sobolev Function Spaces 155
 4. Fourier Transformation of Rapidly Growing Generalized Functions 156
4.1. Functions on the Space Z(C^n) 156
4.2. Fourier Transformation on the Space D (R^n) 157
4.3. Operations on the Space Z (C^n) 158
4.4. Properties of the Fourier Transformation 158
4.5. Analytic Functionals 158
 5. The Paley-Wiener Theory 160
5.1. Fourier Transform of Generalized Functions with Compact Supports 160
5.2. Tempered Distributions with Support in a Cone 160
5.3. Exponentially Growing Distributions Having Support in a Cone 161
 6. Convolution and Fourier Transform 163
Chapter 3. Existence and Uniqueness of Solutions of Differential Equations 164
 1. The Problem of Division 164
1.1. The Problem of Division in Classes of Rapidly Growing Distributions 164
1.2. The Problem of Division in Classes of Exponentially Growing Generalized Functions. The Hrmander Staircase 166
1.3. The Problem of Division in Classes of Tempered Distributions 167
 2. Regularization. The Methods of "Subtraction" and Exit to the Complex Domain and the Riesz Power Method 168
2.1. The Method of Subtraction 169
2.2. The Method of Exit to the Complex Domain 171
2.3. The Riesz Method of Complex Powers 172
 3. Equations in a Convex Cone. An Operational Calculus 173
3.1. Equations in a Cone 173
3.2. An Operational Calculus 175
3.3. Differential-difference Equations on a Semi-axis 177
 4. Propagation of Singularities and Smoothness of Solutions 178
4.1. Characteristics of Differential Equations 178
4.2. Wave Fronts Bicharacteristics and Propagation of Singularities 180
II. Linear Partial Differential Equations with Constant Coefficients 123
 5. Smoothness of Solutions of Elliptic Equations. Hypoellipticity 183
5.1. Smoothness of Generalized Solutions of Elliptic Equations 183
5.2. Hypoelliptic Operators 184
Chapter 4. The Function P^lambda_+ for Polynomials of Second-degree and its Application in the Construction of Fundamental Solutions 186
 1. The Function P^lambda_+ for the Case when P is a Real Linear Function 186
1.1. Analytic Continuation with Respect to lambda 186
1.2. An Application to Bessel Functions 188
 2. The Function P^lambda_+ for the Case when P(x) is a Quadratic Form of the Type (m, n - m) with Real Coefficients 188
2.1. The Case m = n 189
2.2. Application to Decomposition of delta-Function into Plane Waves 190
2.3. The Case 1 <= m <= n - 1 191
2.4. Application to Bessel Functions 193
 3. Invariant Fundamental Solutions of Second-order Equations with Real Coefficients 196
3.1. Analysis of Invariance Properties of the Equation 197
3.2. Determination of the Regular Part of an Invariant Fundamental Solution 198
 4. Regularization of the Formal Fundamental Solution for the Case q = 0 200
4.1. The Case m = 0 or m = n 200
4.2. The Case 1 <= m <= n - 1 201
 5. Regularization of the Fundamental Solution for the Case q not equal 0 204
5.1. The Case 1 <= m <= n - 1 204
5.2. The Case m = 0 or m = n 207
 6. On Singularities of Fundamental Solutions of Second-order Equations with Real Coefficients and with Non-degenerate Quadratic Form 211
Chapter 5. Boundary-value Problems in Half-space 212
 1. Equations with Constant Coefficients in a Half-space 213
1.1. General Solution of Equation (0.1) in a Half-space 213
1.2. Classification of Equations in Half-space 215
 2. Regular Boundary-value Problems in a Half-space in Classes of Bounded Functions 220
2.1. Regular Boundary-value Problems 221
2.2. Examples of Regular Boundary-value Problems 224
 3. Regular Boundary-value Problems in Classes of Exponentially Growing Functions 226
3.1. Definition and Examples 226
3.2. The Cauchy Problem 228
3.3. The Dirichlet Problem for Elliptic Equations 229
 4. Regular Boundary-value Problems in the Class of Functions of Arbitrary Growth 229
 5. Well-posed and Continuous Boundary-value Problems in a Half-space 231
5.1. Well-posed Boundary Value Problems 231
5.2. Continuous Well-posed Boundary-value Problems 232
 6. The Poisson Kernel for the Boundary-value Problem in a Half-space 234
6.1. The Poisson Kernel and the Fundamental Solution of the Boundary-value Problem 234
6.2. The Connection Between the Fundamental Solution of the Cauchy Problem and the Retarded Fundamental Solution of the Operator P(_x) 235
 7. Boundary-value Problems in a Half-space for Non-homogeneous Equations 238
7.1. Non-homogeneous Equations in a Half-space 238
7.2. Boundary-value Problems for Non-homogeneous Equations 240
Chapter 6. Sharp and Diffusion Fronts of Hyperbolic Equations 240
 1. Basic Notions 241
 2. The Petrovskij Criterion 244
 3. The Local Petrovskij Criterion 246
 4. Geometry of Lacunae Near Concrete Singularities of Fronts 247
 5. Equations with Variable Coefficients 250
Bibliographical Comments 250
References 251
Author Index 257
Subject Index 261
END
