Analytic Partial Differential Equations (eBook)

François Treves (Autor)

eBook Download: PDF

2022 | 2022
XIII, 1228 Seiten
Springer International Publishing (Verlag)
978-3-030-94055-3 (ISBN)

Infos

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 Fourier-Bros-Iagolnitzer (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 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 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 (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 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.

François Treves has worked in the field of partial differential equations for 65 years, publishing 17 books and over 150 papers. His well-known contributions to the field include his work with Louis Nirenberg on the local solvability of linear PDEs, his contributions to the theory of locally integrable structures (presented in his book Hypo-analytic Manifolds, Princeton University Press, 1991), his results on the analyticity of the solutions of sum-of-square PDEs (Tartakoff-Treves theorem), his characterization of the conserved quantities of the KDV equation, and his extension of the Ovsyannikov approach to the Cauchy-Kovalevskaya theorem and beyond.

Produkt-Details

• Erscheint lt. Verlag: 26.4.2022
• Reihe/Serie: Grundlehren der mathematischen Wissenschaften
• Reihe/Serie: Grundlehren der mathematischen Wissenschaften
• Zusatzinfo: XIII, 1228 p.
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Schlagworte: analytic partial differential equations • analytic wave-front set • Cauchy-Kovalevskaya Theorem • differential complexes • Distribution • eikonal equations • FBI transform • Fourier integral operators • hyperfunctions • Lojaciewicz inequality • microlocal analysis • Nagano foliations • Ovsyannikov analyticity • Partial differential equations • partial differential equations of principal type • phase-functions • pseudodifferential operators • regularity of hyperfunction solutions • Stratification
• ISBN-10: 3-030-94055-1 / 3030940551
• ISBN-13: 978-3-030-94055-3 / 9783030940553