Overview
- A self-contained introduction to the analysis of solutions of partial differential equations on manifolds
- Introduces tools of microlocal analysis such as the FBI transform and pseudodifferential operators
- Includes background on sheaf theory, differential and symplectic geometry, and stratification theory
Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 359)
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Table of contents (27 chapters)
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Front Matter
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Distributions and Analyticity in Euclidean Space
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Front Matter
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Hyperfunctions in Euclidean Space
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Front Matter
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Stratification of Analytic Varieties and Division of Distributions by Analytic Functions
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Front Matter
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Keywords
- partial differential equations
- analytic partial differential equations
- distribution
- microlocal analysis
- hyperfunctions
- Ovsyannikov analyticity
- FBI transform
- analytic wave-front set
- pseudodifferential operators
- Fourier integral operators
- partial differential equations of principal type
- differential complexes
- phase-functions
- Cauchy-Kovalevskaya Theorem
- Lojaciewicz inequality
- stratification
- Nagano foliations
- eikonal equations
- regularity of hyperfunction solutions
About this book
The book begins by establishing the fundamental properties of analytic partial differential equations, starting with the Cauchy–Kovalevskaya 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 Hamilton–Jacobi solutions of involutive systems of eikonalequations. 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 (à la Sjöstrand). 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 equations—a 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 analyticvarieties 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.
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Analytic Partial Differential Equations
Authors: François Treves
Series Title: Grundlehren der mathematischen Wissenschaften
DOI: https://doi.org/10.1007/978-3-030-94055-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-030-94054-6Published: 27 April 2022
eBook ISBN: 978-3-030-94055-3Published: 26 April 2022
Series ISSN: 0072-7830
Series E-ISSN: 2196-9701
Edition Number: 1
Number of Pages: XIII, 1228
Topics: Global Analysis and Analysis on Manifolds, Analysis, Fourier Analysis