偏微分方程:Ⅳ:微局部分析和双曲型方程 目录

PrefaceChapter 1.Microlocal Properties of Distributions2.Wave Front of Distribution.Its Functorial Properties2.1.Definition ofthe Wave Front2.2.Localization ofWave Front2.3.Wave Front and Singularities of One-Dimensional2.4.Wave Fronts of Pushforwards and Pullbacks of a3.Wave Front and Operations on Distributions3.1 The Trace of a Distribution.Product of Distnritbiaul Eiuation3.2.The Wave Front of the Solution of a Differential Eqution3.3.Wave Fronts and Integral OperatorsChapter 2.Pseudodifferential Operators1.Algebra ofPseudodifferential Operators1.1.Singular Integral Operators1.2.The Symbol1.3.Boundedness of Pseudodifferential Operators1.4.Composition of Pseudodifferential Operators1.5.The Formally Adjoint Operator1.6.Pseudolocality.Microlocality1.7.Elliptic Operators1.8.Garding'S Inequality1.9.Extension 0f the Class of Pseudodifferential Operators2.Invariance of the Principal SymboJ Under Canonical Transformations2.1.Invariance Under the Change ofVariables.2.2 The Subprincipal Symbol2.3.Canonical Transformations2.4.An Inverse Theorem3.Canonical Forms ofthe Symbol3.1.Simple Characteristic Points3.2.Double Characteristics3.3.The Complex-alued Symbol3.4.The Canonical Form of the Symbol in a Neighbourhood of the Boundary.4.Various Classes of Pseudodifferential Operators4.1.The Lm/pδClasses4.2.The Lm/φ,φ Classes4.3.The Weyl Operators5.Complex Powers ofElliptic Operators5.1.The Definition ofComplex Powers.5.2.Thc Construction of the Symbol for the Operator Az5.3.The Construction of the Kernel of the Operator Az5.4.The ξ-Function ofan Elliptic Operator5.5.The Asymptotics of the Spectral Function and Eigenvalues5.6.Complex Powers of an Elliptic Operator with Boundary Conditions6.Pseudodifferential Operators in IRn and Quantization6.1.The Analogy Between the Microlocal Analysis and the Quantization6.2.Pseudodifierential 0perators in RnChapter 3.Fourier Integral Operators1.The Parametrix of the Cauchy Problem for Hyperbolic Equations1.1.The Cauchy Problem for the Wave Equation1.2.The Cauchy Problem for the Hyperbolic Equation of an Arbitrary 0rder.1.3.The Method of Stationary Phase2.The Maslov Canonical Operator2.1.The MaslOV Index2.2.Pre.canonieal Operator2.3.The Canonical Operator2.4.Some Applications.3.Fourier Integral Operators3.1.The Oscillatory Integrals3.2.The Local Definition of the Fourier Integral Operator3.3. The Equivalence of Phase Functions3.4. The Connection with the Lagrange Manifold3.5. The Global Definition of the Fourier Distribution3.6. The Global Fourier Integral Operators4. The Calculus of Fourier Integral Operators4.1. The Adjoint Operator4.2. The Composition of Fourier Integral Operators4.3. The Boundedness in L25. The Image of the Wave Front Under the Action of a Fourier Integral Operator5.1. The Singularities of Fourier Integrals5.2. The Wave Front of the Fourier Integral5.3. The Action of the Fourier Integral Operator on Wave Fronts6. Fourier Integral Operators with Complex Phase Functions6.1. The Complex Phase6.2. Almost Analytic Continuation6.3. The Formula for Stationary Complex Phase6.4. The Lagrange Manifold6.5. The Equivalence of Phase Functions6.6. The Principal Symbol6.7. Fourier Integral Operators with Complex Phase Functions 6.8. Some ApplicationsChapter 4. The Propagation of Singularities1. The Regularity of the Solution at Non-characteristic Points1.1. The Microlocal Smoothness1.2. The Smoothness of Solution at a Non-characteristic Point2. Theorems on Removable Singularities2.1. Removable Singularities in the Right-Hand Sides of Equations2.2. Removable Singularities in Boundary Conditions3. The Propagation of Singularities for Solutions of Equations of Real Principal Type3.1. The Definition and an Example3.2. A Theorem of H6rmander3.3. Local Solvability3.4. Semiglobal Solvability4. The Propagation of Singularities for Principal Type Equations with a Complex Symbol4.1. An Example4.2. The Fixed Singularity4.3. A Special Case4.4. The Propagation of Singularities in the Case of a Complex Symbol of the General Form5. Multiple Characteristics5.1. Non-involutive Double Characteristics5.2. The Levi Condition5.3. Operators Having Characteristics of Constant Multiplicity .5.4. Operators with Involutive Multiple Characteristics5.5. The Schrrdinger OperatorChapter 5. Solvability of (Pseudo)Differential Equations1. Examples1.1. Lewy's Example1.2. Mizohata's Equation1.3. Other Examples2. Necessary Conditions for Local Solvability2.1. Hrrmander's Theorem2.2. The Zero of Finite Order2.3. The Zero of Infinite Order2.4. Multiple Characteristics3. Sufficient Conditions for Local Solvability3.1. Operators of Real Principal Type3.2. Operators of Principal Type3.3. Operators with Multiple CharacteristicsChapter 6. Smoothness of Solutions of Differential Equations1. Hypoelliptic Operators1.1.Definition and Examples1.2.Hypoelliptic Differential Operators with Constant Coefficients1.3. The Gevrey Classes1.4.Partially HypoeUiptic Operators1.5.HypoeUiptic Equations in Convolutions1.6.Hypoelliptic Operators of Constant Strength1.7. Hypoelliptic Differential Operators with Variable Coefficients1.8.Pseudodifferential Hypoelliptic Operators1.9. Degenerate Elliptic Operators1.10. Partial Hypoellipticity of Degenerate Elliptic Operators1.11. Double Characteristics1.12. Hypoelliptic Operators on the Real Line2. Subeiliptic Operators2.1. Definition and Simplest Properties2.2. Estimates for First-Order Differential Operators with Polynomial Coefficients2.3. Algebraic Conditions

偏微分方程:Ⅳ:微局部分析和双曲型方程 内容简介

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偏微分方程:Ⅳ:微局部分析和双曲型方程 节选

This volume of the Encyclopaedia contains two contributions．In the first Yu.V.Egorov gives an accomnt of microlocal analysis as a tool for investigating partial differemial equations．This 113ethod has become increasingly important in the theory．of Hamiltonian systems in recent years． The second survey written by V.Ya．1vrii treats linear hyperbolic equations and systems．The author states necessary and sufficiient conditions for C∞-and L2-well-posedness and he studies the analogous pmhlem in the comext ofGevrey 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 coyer recent research in two important fields,which before was scattered in numerous joumals．The book will hence be of immense value to graduate students and researchers in partial differential equationS and theoretical physics。