說明 
xiii, 1228 pages ; 24 cm 

text txt rdacontent 

unmediated n rdamedia 

volume nc rdacarrier 
系列 
Grundlehren der mathematischen Wissenschaften, 00727830 ; Volume 359


Grundlehren der mathematischen Wissenschaften ; 359

附註 
Includes bibliographical references and indexes 

This book provides a coherent, selfcontained introduction to central topics of Analytic Partial Differential Equations in the natural geometric setting. The main themes are the analysis in phasespace of analytic PDEs and the FourierBrosIagolnitzer (FBI) transform of distributions and hyperfunctions, with application to existence and regularity questions. The book begins by establishing the fundamental properties of analytic partial differential equations, starting with the CauchyKovalevskaya theorem, before presenting an integrated overview of the approach to hyperfunctions via analytic functionals, first in Euclidean space and, once the geometric background has been laid out, on analytic manifolds. Further topics include the proof of the Lojaciewicz inequality and the division of distributions by analytic functions, a detailed description of the Frobenius and Nagano foliations, and the HamiltonJacobi solutions of involutive systems of eikonal equations. The reader then enters the realm of microlocal analysis, through pseudodifferential calculus, introduced at a basic level, followed by Fourier integral operators, including those with complex phasefunctions (a la Sjostrand). This culminates in an indepth discussion of the existence and regularity of (distribution or hyperfunction) solutions of analytic differential (and later, pseudodifferential) equations of principal type, exemplifying the usefulness of all the concepts and tools previously introduced. The final three chapters touch on the possible extension of the results to systems of over (or under) determined systems of these equationsa cornucopia of open problems. This book provides a unified presentation of a wealth of material that was previously restricted to research articles. In contrast to existing monographs, the approach of the book is analytic rather than algebraic, and tools such as sheaf cohomology, stratification theory of analytic varieties and symplectic geometry are used sparingly and introduced as required. The first half of the book is mainly pedagogical in intent, accessible to advanced graduate students and postdocs, while the second, more specialized part is intended as a reference for researchers 

Distributions and Analyticity in Euclidean Space  Functions and Differential Operators in Euclidean Space  Basic Notation and Terminology  Smooth, Realanalytic, Holomorphic Functions  Differential Operators with Smooth Coefficients  Distributions in Euclidean Space  Basics on Distributions in Euclidean Space  Sobolev Spaces  Distribution Kernels  Fundamental Solutions, Parametrix, Hypoelliptic PDOs  Analytic Tools in Distribution Theory  Analytic Parametrices, Analytic Hypoellipticity  Ehrenpreisõtoffs and Analytic Regularity of Distributions  Distribution Boundary Values of Holomorphic Functions  The FBI Transform of Distributions : An Introduction  The Analytic WaveFront Set of a Distribution  Analyticity of Solutions of Linear PDEs : Basic Results  Analyticity of Solutions of Elliptic Linear PDEs  Degenerate Elliptic Equations : Influence of Lower Order Terms A Generalization of the Harmonic Oscillator  Appendix : Hermite's Functions and the Schwartz Space  The CauchyKovalevskaya Theorem  A Nonlinear Ovsyannikov Theorem  Application : the Nonlinear CauchyKovalevskaya Theorem  Applications to Linear PDE  Application to Integrodifferential Cauchy Problems  Hyperfunctions in Euclidean Space  Analytic Functionals in Euclidean Space  Analytic Functionals in Complex Domains  Analytic Functionals in Cn  Analytic Functionals in Rn as Cohomology Classes  Hyperfunctions in Euclidean Space  The Sheaf of Hyperfunctions in Euclidean Space  Boundary values of holomorphic functions in wedges  The FBI Transform of Analytic Functionals  Analytic Wavefront Set of a Hyperfunction  Edge of the Wedge  Microfunctions in Euclidean space  Hyperdifferential Operators  Action on Holomorphic Functions and on Hyperfunctions  Local Representation of Hyperfunctions Elliptic Hyperdifferential Operators  Solvability of Constant Coefficients Hyperdifferential Equations  Geometric Background  Elements of Differential Geometry  Regular Manifolds  Fibre Bundles, Vector Bundles  Tangent and Cotangent Bundles of a Manifold  Differential Complexes and Grassman Algebras  A Primer on Sheaf Cohomology  Basics on Sheaf Cohomology  Fine Sheaves and Fine Resolutions  Relative Sheaf Cohomology  Edge of the Wedge in (Co)homological Terms  Distributions and Hyperfunctions on a Manifold  Distributions and Currents on a Manifold  Plurisubharmonic functions and pseudoconvex domains  Hyperfunctions and Microfunctions in an Analytic Manifold  Lie Algebras of Vector Fields  The Lie Algebra of Smooth Vector Fields  Integral Manifolds : FrobeniusԨeorem  Local Flow of a Regular Vector Field  Foliations Defined By Analytic Vector Fieldsresentation of Hyperfunctions Systems of Vector Fields Generating Special Lie Algebras  Elements of Symplectic Geometry  Elements of Symplectic Algebra  The Metaplectic Group  Symplectic Manifolds  Involutive Systems of Functions of Principal Type  Real and Imaginary Symplectic Structures in C2n  Real and Imaginary Symplectic Structures on Complex Manifolds  Stratification of Analytic Varieties and Division of Distributions by Analytic Functions  Analytic Stratifications  Analytic Stratifications and Stratifiable Sets  Analytic Subvarieties  The Weierstrass Theorems  Local Partitions of a Complex Hypersurface  Local Stratifications of a RealAnalytic Variety  Semianalytic Sets  Division of Distributions by Analytic Functions  The Lojasiewicz Inequality  Division of Distributions by Analytic Functions  Desingularization and Applications  Appendix  Analytic Pseudodifferential Operators and Fourier Integral Operators Elementary Pseudodifferential Calculus in the ... Class  Standard Pseudodifferential Operators  Symbolic Calculus  Classical symbols and classical pseudodifferential operators  The Weyl Calculus in Euclidean Space  Analytic Pseudodifferential Calculus  Analytic Pseudodifferential Operators  Symbolic Calculus  Analytic Microlocalization In Distribution Theory  Action on Singularity Hyperfunctions  Microdifferential Operators  Fourier Integral Operators  Fourier Distribution Kernels in Euclidean Space  The Lagrangian Manifold Associated to a Phasefunction  Fourier Integral Operators : Basics  Reduction of the Fiber Variables  Composition and Continuity of Fourier Integral Operators  Globally Defined Fourier Integral Operators  Principles of Analytic Fourier Integral Operators  Appendix : Stationary Phase Formal Expansion  Complex Microlocal Analysis  Classical Analytic Formalismtors Formal Analytic Series  Classical Analytic Differential Operators of Infinite Order  The Complex Stationary Phase Formula  Symbolic Calculus and the KdV Hierarchy  Germ Fourier Integral Operators in Complex Space  Analytic Symbols  Contours and Function Spaces  Sj诳trand Pairs  Germ Fourierlike Transforms  Sj诳trand Triads and Germ Fourier Integral Operators  Germ Pseudodifferential Operators in Complex Space  Germ Pseudodifferential Operators  Classical Germ Pseudodifferential Operators  Action on distributions  Action on Hyperfunctions and Microfunctions  Germ FBI Transforms  Germ FBI Transforms  Germ FBI Transforms of Distributions  The Equivalence Theorem for Distributions  Analytic Pseudodifferential Operators of Principal Type  Analytic PDEs of Principal Type : Local Solvability  Pseudodifferential Operators of Principal Type  Local Solvability of Analytic PDEs of Principal Type Analytic PDEs of Principal Type : Regularity of the Solutions  A New Concept : Subellipticity  Statement of the Main Theorem  Hypoellipticity Implies (Q)  Property (Q) Implies Subellipticity  Analytic Hypoellipticity Implies (Q)  Property (Q) Implies Analytic Hypoellipticity  The ... Situation  Propagation of Analytic Singularities  Appendix : Properties of Real Polynomials in a Single Variable  Appendix : Analytic Estimates of Exponential Amplitudes  Solvability of Constant Vector Fields of Type (1,0)  CConvexity and Global Solvability  Local Solvability at the Boundary : First Steps  Local Solvability at the Boundary : Final Characterization  The Differential Complex : Generalities  Appendix : Minima of Families of Plurisubharmonic Functions  Pseudodifferential Solvability and Property (...)  Solvability : the Difference between Differential and Pseudodifferential  Property  Principal Type Microlocal Solvability in Distributions  Pseudodifferential Complexes in Tube Structures  Pseudodifferential Complexes of Principal Type  Tube Pseudodifferential Complexes  Phasefunction and Amplitude  Approximate Homotopy Formulas  Homotopy Formulas  Poincar⥠Lemmas  References  Notation Index  Index 
鏈接 
Online version: Treves, Francois, 1930 Analytic partial differential equations.
Cham : Springer, [2022] 9783030940553
(OCoLC)1313387532

主題 
Differential equations, Partial


ⅱuations aux d⥲iv⥥s partielles


Differential equations, Partial. fast (OCoLC)fst00893484

