This volume of the Encyclopaedia contains two contributions. In the first Yu. V. Egorov gives an account of microlocal analysis as a tool for investigating partial differential equations. This method has become increasingly important in the theory of Hamiltonian systems in recent years. The second survey written by V. Ya. lvrii treats linear hyperbolic equations and systems. The author states necessary and sufficient conditions for C∞-and L2-well-posedness and he studies the analogous problem in the context of Gevrey classes. He also describes the latest results in the theory of mixed problems for hyperbolic operators and concludes with a list of unsolved problems. Both parts cover recent research in two important fields, which before was scattered in numerous journals. The book will hence be of immense value to graduate students and researchers in partial differential equations and theoretical physics.
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目录
Ⅰ. Microlocal Analysis Preface Chapter 1 Microlocal Properties of Distributions §1. Microlocalization §2. Wave Front of Distribution. Its Functorial Properties 2.1. Definition of the Wave Front 2.2. Localization of Wave Front 2.3. Wave Front and Singularities of One-Dimensional Distributions 2.4. Wave Fronts of Pushforwards and Pullbacks of a Distribution §3. Wave Front and Operations on Distributions 3.1. The Trace of a Distribution. Product of Distributions 3.2. The Wave Front of the Solution of a Differential Equation 3.3. Wave Fronts and Integral Operators Chapter 2 Pseudodifferential Operators §1. Algebra of Pseudodifferential Operators 1.1. Singular Integral Operators 1.2. The symbol 1.3. Boundedness of Pseudodifferential Operators 1.4. Composition of Pseudodifferential Operators 1.5. The Formally Adjoint Operator 1.6. Pseudolocality. Microlocality 1.7. Elliptic Operators 1.8. Girding’s Inequality 1.9. Extension of the Class of Pseudodifferential Operators §2. Invariance of the Principal Symbol Under Canonical Transformations 2.1. Invariance Under the Change of Variables 2.2. The Subprincipal Symbol 2.3. Canonical Transformations 2.4. An Inverse Theorem §3 . Canonical Forms of the Symbol 3.1. Simple Characteristic Points 3.2. Double Characteristics 3.3. The Complex-Valued Symbol 3.4. The Canonical Form of the Symbol in a Neighbourhood of the Boundary §4. Various Classes of Pseudodifferential Operators 4.1. The Lmp,δ Classes 4.2. The Lφp,φ Classes 4.3. The Weyl Operators §5. Complex Powers of Elliptic Operators 5.1. The Definition of Complex Powers 5.2. The Construction of the Symbol for the Operator Az 5.3. The Construction of the Kernel of the Operator Az 5.4. The ξ-Function of an Elliptic Operator 5.5. The Asymptotics of the Spectral Function and Eigenvalues 5.6. Complex Powers of an Elliptic Operator with Boundary Conditions §6. Pseudodifferential Operators in IRn and Quantization 6.1. The Analogy Between the Microlocal Analysis and the Quantization 6.2. Pseudodifferential Operators in IRn Chapter 3 Fourier Integral Operators §1. The Parametrix of the Cauchy Problem for Hyperbolic Equations 1.1. The Cauchy Problem for the Wave Equation 1.2. The Cauchy Problem for the Hyperbolic Equation of an Arbitrary Order 1.3. The Method of Stationary Phase §2. The Maslov Canonical Operator 2.1. The Maslov Index 2.2. Pre-canonical Operator 2.3. The Canonical Operator 2.4. Some Applications §3. Fourier Integral Operators 3.1. The Oscillatory Integrals 3.2. The Local Definition of the Fourier Integral Operator 3.3. The Equivalence of Phase Functions 3.4. The Connection with the Lagrange Manifold 3.5. The Global Definition of the Fourier Distribution 3.6. The Global Fourier Integral Operators §4. The Calculus of Fourier Integral Operators 4.1. The Adjoint Operator 4.2. The Composition of Fourier Integral Operators 4.3. The Boundedness in L2 §5. The Image of the Wave Front Under the Action of a Fourier Integral Operator 5.1. The Singularities of Fourier Integrals 5.2. The Wave Front of the Fourier Integral 5.3. The Action of the Fourier Integral Operator on Wave Fronts §6. Fourier Integral Operators with Complex Phase Functions 6.1. The Complex Phase 6.2. Almost Analytic Continuation 6.3. The Formula for Stationary Complex Phase 6.4. The Lagrange Manifold 6.5. The Equivalence of Phase Functions 6.6. The Principal Symbol 6.7. Fourier Integral Operators with Complex Phase Functions 6.8. Some Applications Chapter 4 The Propagation of Singularities §1. The Regularity of the Solution at Non-characteristic Points 1.1. The Microlocal Smoothness 1.2. The Smoothness of Solution at a Non-characteristic Point §2. Theorems on Removable Singularities 2.1. Removable Singularities in the Right-Hand Sides of Equations 2.2. Removable Singularities in Boundary Conditions §3. The Propagation of Singularities for Solutions of Equations of Real Principal Type 3.1. The Definition and an Example 3.2. A Theorem of Hormander 3.3. Local Solvability 3.4. Semiglobal Solvability §4. The Propagation of Singularities for Principal Type Equations with a Complex Symbol 4.1. An Example 4.2. The Fixed Singularity 4.3. A Special Case 4.4. The Propagation of Singularities in the Case of a Complex Symbol of the General Form §5. Multiple Characteristics 5.1. Non-involutive Double Characteristics 5.2. The Levi Condition 5.3. Operators Having Characteristics of Constant Multiplicity 5.4. Operators with Involutive Multiple Characteristics 5.5. The Schrodinger Operator Chapter 5 Solvability of (Pseud0)Differential Equations §1. Examples 1.1. Lewy’s Example 1.2. Mizohata’s Equation 1.3. Other Examples §2. Necessary Conditions for Local Solvability 2.1. Hormander’s Theorem 2.2. The Zero of Finite Order 2.3. The Zero of Infinite Order 2.4. Multiple Characteristics §3. Sufficient Conditions for Local Solvability 3.1. Operators of Real Principal Type 3.2. Operators of Principal Type 3.3. Operators with Multiple Characteristics Chapter 6 Smoothness of Solutions of Differential Equations §1. Hypoelliptic Operators 1.1. Definition and Examples 1.2. Hypoelliptic Differential Operators with Constant Coefficients 1.3. The Gevrey Classes 1.4. Partially Hypoelliptic Operators 1.5. Hypoelliptic Equations in Convolutions 1.6. Hypoelliptic Operators of Constant Strength 1.7. Hypoelliptic Differential Operators with Variable Coefficients 1.8. Pseudodifferential Hypoelliptic Operators 1.9. Degenerate Elliptic Operators 1.10. Partial Hypoellipticity of Degenerate Elliptic Operators 1.11. Double Characteristics 1.12. Hypoelliptic Operators on the Real Line §2. Subelliptic Operators 2.1. Definition and Simplest Properties 2.2. Estimates for First-Order Differential Operators with Polynomial Coefficients 2.3. Algebraic Conditions §3. Hypoelliptic Differential Operators of Second Order 3.1. The Sum of the Squares 3.2. A Necessary Condition for Hypoellipticity 3.3. Operators with a Non-negative Quadratic Form §4. Analytic Hypoellipticity 4.1. Elliptic Operators 4.2. The Analytic Wave Front 4.3. Analytic Pseudodifferential Operators 4.4. Necessary Conditions for Analytic Hypoellipticity 4.5. Differential Equation of the Second Order 4.6. The Gevrey Classes 4.7. Generalized Analytic Hypoellipticity Chapter 7 Transformation of Boundary-Value Problems §1. The Transmission Property 1.1. Operators in a Half-Space 1.2. The Transmission Property 1.3. Application to the Study of Lacunae §2. Distributions on a Manifold with Boundary 2.1. The Distribution Spaces 2.2. Contracted Cotangent Bundle §3. Completely Characteristic Operators 3.1. Pseudodifferential Operators and their Kernels 3.2. The Transmission Property 3.3. Completely Characteristic Operators 3.4. The Boundary Wave Front §4. Canonical Boundary Transformation 4.1. The Generating Function 4.2. The Operator of Principal Type 4.3. The Differential Operator of Second Order §5. Fourier Integral Operators 5.1. The Generating Function of the Canonical Boundary Transformation 5.2. The Fourier Integral Operator Chapter 8 Hyperfunctions §1. Analytic Functionals 1.1. Definition and the Basic Properties 1.2. Operations on Analytic Functionals §2. The Space of Hyperfunctions 2.1. Definition and the Basic Properties 2.2. The Analytic Wave Front of a Hyperfunction 2.3. Boundary Values of a Hyperfunction §3. Solutions of Differential Equations 3.1. The Cauchy Problem 3.2. The Analytic Wave Front §4 . Sheaf of Microfunctions 4.1. Traces of Holomorphic Functions 4.2. The Definition of a Sheaf of Microfunctions 4.3. Pseudodifferential Operators 4.4. Fourier Integral Operators References Ⅱ. Linear Hyperbolic Equations Introduction Chapter 1 The Cauchy Problem §1. C∞-well-posedness of the Cauchy Problem 1.1. Basic Definitions and Notation 1.2. Necessity of Hyperbolicity. L2-well-posedness of the Cauchy Problem 1.3. Operators with Constant Coefficients. Algebraic Properties of Hyperbolic Polynomials 1.4. Regularly and Completely Regularly Hyperbolic Operators 1.5. Operators with Characteristics of Constant Multiplicity 1.6. Irregularly Hyperbolic Operators of General Form 1.7. Necessary Conditions for the Cauchy Problem to be Well Posed 1.8. Degenerate Hyperbolic Equations 1.9. Second-Order Equations in Two Variables 1.10. Systems with Characteristic Roots of Constant Multiplicity 1.1 1 . Necessary Conditions for Regular Hyperbolicity of First-Order Systems in Two Variables §2. Well-posedness of the Cauchy Problem in Gevrey Classes 2.1. The Main Definitions 2.2. Necessity and Sufficiency of Hyperbolicity 2.3. Operators with Constant Coefficients 2.4. Operators with Characteristics of Constant Multiplicity 2.5. Necessary Conditions for Well-posedness in the Gevrey Classes 2.6. Examples §3. Propagation of Singularities 3.1. Propagation of Cm-Singularities 3.2. The Geometry of the Propagation of Singularities 3.3. The Construction of a Parametrix 3.4. Propagation of Analytic Singularities and Gevrey Singularities Chapter 2 Mixed Problems for Hyperbolic Operators §1. Well-posedness of Mixed Problems 1.1. Preliminary Remarks 1.2. Operators with Constant Coefficients 1.3. Strong L2-well-posedness of the Mixed Problem(for Equations) 1.4. Strong L2-well-posedness of the Mixed Problem for Systems of First Order 1.5. Necessary Conditions for Cw-well-posedness of the Mixed Problem 1.6. Weak L2-well-posedness of the Mixed Problem 1.7. C∞-well Posed Mixed Problem for Strictly Hyperbolic Equations of Second Order 1.8. Mixed Problem in the Classes of Analytic Functions §2. Propagation of Cm-Singularities 2.1. The Wave Fronts 2.2. Propagation of Singularities of Solutions to Dissipative Boundary-Value Problems for Symmetric Hyperbolic Systems 2.3. The Geometry of the Propagation of Singularities 2.4. The Propagation of Singularities of Solutions to Strictly Dissipative Boundary-Value Problems for Symmetric 2.5. The Geometry of the Propagation of Singularities(the Concluding Part) 2.6. The Propagation of Singularities of Solutions to Non-classical Problems §3. The Propagation of Analytic Singularities 3.1. The General Theory 3.2. The Wave Equation Problems Bibliographical Remarks A Survey of Recent Results References Author Index Subject Index
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