| 說明 |
xiii, 1228 pages ; 24 cm |
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text txt rdacontent |
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unmediated n rdamedia |
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volume nc rdacarrier |
| 系列 |
Grundlehren der mathematischen Wissenschaften, 0072-7830 ; Volume 359
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Grundlehren der mathematischen Wissenschaften ; 359
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| 附註 |
Includes bibliographical references and indexes |
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This book provides a coherent, self-contained introduction to central topics of Analytic Partial Differential Equations in the natural geometric setting. The main themes are the analysis in phase-space 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 phase-functions (a la Sjostrand). This culminates in an in-depth 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 |
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Distributions and Analyticity in Euclidean Space -- Functions and Differential Operators in Euclidean Space -- Basic Notation and Terminology -- Smooth, Real-analytic, 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 Wave-Front 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 Cauchy-Kovalevskaya Theorem -- A Nonlinear Ovsyannikov Theorem -- Application : the Nonlinear Cauchy-Kovalevskaya 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 Wave-front 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 Real-Analytic 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 Phase-function -- 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 Fourier-like 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) -- C-Convexity 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 -- Phase-function 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
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| 主題 |
Differential equations, Partial
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ⅱuations aux d⥲iv⥥s partielles
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Differential equations, Partial. fast (OCoLC)fst00893484
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