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Nonelliptic Partial Differential Equations

Analytic Hypoellipticity and the Courage to Localize High Powers of T

  • Book
  • © 2011

Overview

Authors:
  1. David S. Tartakoff

    1. Chicago, USA

    View author publications

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  • The exposition is generous and relaxed, allowing the reader to come to terms to the technique at their own pace. The main difficulty, localization, is approached directly from the beginning with simple examples
  • Numerous applications are included
  • There is no similar book
  • There are other techniques for proving analytic hypoellipticity, but each has its own difficulties. While this is elementary but not simple, once the few basic formulas are established the rest is combinatorial in nature, and not conceptually difficult
  • Includes supplementary material: sn.pub/extras

Part of the book series: Developments in Mathematics (DEVM, volume 22)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-viii
    Download chapter PDF

  2. What This Book Is and Is Not

    • David S. Tartakoff
    Pages 1-3

  3. Brief Introduction

    • David S. Tartakoff
    Pages 5-6

  4. Overview of Proofs

    • David S. Tartakoff
    Pages 7-32

  5. Full Proof for the Heisenberg Group

    • David S. Tartakoff
    Pages 33-39

  6. Coefficients

    • David S. Tartakoff
    Pages 41-64

  7. Pseudodifferential Problems

    • David S. Tartakoff
    Pages 65-70

  8. General Sums of Squares of Real Vector Fields

    • David S. Tartakoff
    Pages 71-79

  9. The \(\overline{\partial }\)-Neumann Problem and the Boundary Laplacian on Strictly Pseudoconvex Domains

    • David S. Tartakoff
    Pages 81-89

  10. Symmetric Degeneracies

    • David S. Tartakoff
    Pages 91-98

  11. Details of the Previous Chapter

    • David S. Tartakoff
    Pages 99-130

  12. Nonsymplectic Strata and Germ Analytic Hypoellipticity

    • David S. Tartakoff
    Pages 131-151

  13. Operators of Kohn Type That Lose Derivatives

    • David S. Tartakoff
    Pages 153-157

  14. Nonlinear Problems

    • David S. Tartakoff
    Pages 159-189

  15. Treves’ Approach

    • David S. Tartakoff
    Pages 191-193

  16. Appendix

    • David S. Tartakoff
    Pages 195-198

  17. Back Matter

    Pages 199-203
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Keywords

About this book

This book provides a very readable description of a technique, developed by the author years ago but as current as ever, for proving that solutions to certain (non-elliptic) partial differential equations only have real analytic solutions when the data are real analytic (locally). The technique is completely elementary but relies on a construction, a kind of a non-commutative power series, to localize the analysis of high powers of derivatives in the so-called bad direction. It is hoped that this work will permit a far greater audience of researchers to come to a deep understanding of this technique and its power and flexibility.

Reviews

From the reviews:

“The present book deals with the analytic and Gevrey local hypoellipticity of certain nonelliptic partial differential operators. … this nice book is mostly addressed to Ph.D. students and researchers in harmonic analysis and partial differential equations, the reader being supposed to be familiar with the basic facts of pseudodifferential calculus and several complex variables. It represents the first presentation, in book form, of the challenging and still open problem of analytic and Gevrey hypoellipticity of sum-of-squares operators.” (Fabio Nicola, Mathematical Reviews, Issue 2012 h)

Authors and Affiliations

  • Chicago, USA

    David S. Tartakoff

Bibliographic Information

  • Book Title: Nonelliptic Partial Differential Equations

  • Book Subtitle: Analytic Hypoellipticity and the Courage to Localize High Powers of T

  • Authors: David S. Tartakoff

  • Series Title: Developments in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4419-9813-2

  • Publisher: Springer New York, NY

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Science+Business Media, LLC 2011

  • Hardcover ISBN: 978-1-4419-9812-5Published: 26 July 2011

  • Softcover ISBN: 978-1-4614-2969-2Published: 15 August 2013

  • eBook ISBN: 978-1-4419-9813-2Published: 26 July 2011

  • Series ISSN: 1389-2177

  • Series E-ISSN: 2197-795X

  • Edition Number: 1

  • Number of Pages: VIII, 203

  • Topics: Partial Differential Equations, Analysis

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