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Distributions, Partial Differential Equations, and Harmonic Analysis

  • Textbook
  • © 2018

Overview

Authors:
  1. Dorina Mitrea

    1. Department of Mathematics, University of Missouri, Columbia, USA

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  • Pursues the development of distribution theory simultaneously with relevant incursions into partial differential equations and harmonic analysis
  • Written in a concise, rigorous, and largely self-contained form with a balanced treatment of the topics involved
  • Equips readers to handle subsequent courses dealing with allied topics, including: Harmonic Analysis, Partial Differential Equations, Boundary Integral Methods, Sobolev Spaces, Pseudodifferential Operators

Part of the book series: Universitext (UTX)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xxiii
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  2. Weak Derivatives

    • Dorina Mitrea
    Pages 1-16

  3. The Space \(\mathcal {D} '(\varOmega )\) of Distributions

    • Dorina Mitrea
    Pages 17-95

  4. The Schwartz Space and the Fourier Transform

    • Dorina Mitrea
    Pages 97-115

  5. The Space of Tempered Distributions

    • Dorina Mitrea
    Pages 117-202

  6. The Concept of Fundamental Solution

    • Dorina Mitrea
    Pages 203-214

  7. Hypoelliptic Operators

    • Dorina Mitrea
    Pages 215-231

  8. The Laplacian and Related Operators

    • Dorina Mitrea
    Pages 233-320

  9. The Heat Operator and Related Versions

    • Dorina Mitrea
    Pages 321-332

  10. The Wave Operator

    • Dorina Mitrea
    Pages 333-353

  11. The Lamé and Stokes Operators

    • Dorina Mitrea
    Pages 355-386

  12. More on Fundamental Solutions for Systems

    • Dorina Mitrea
    Pages 387-424

  13. Sobolev Spaces

    • Dorina Mitrea
    Pages 425-496

  14. Solutions to Selected Exercises

    • Dorina Mitrea
    Pages 497-541

  15. Appendix

    • Dorina Mitrea
    Pages 543-590

  16. Back Matter

    Pages 591-600
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Keywords

About this book

The aim of this book is to offer, in a concise, rigorous, and largely self-contained manner, a rapid introduction to the theory of distributions and its applications to partial differential equations and harmonic analysis. The book is written in a format suitable for a graduate course spanning either over one-semester, when the focus is primarily on the foundational aspects, or over a two-semester period that allows for the proper amount of time to cover all intended applications as well. It presents a balanced treatment of the topics involved, and contains a large number of exercises (upwards of two hundred, more than half of which are accompanied by solutions), which have been carefully chosen to amplify the effect, and substantiate the power and scope, of the theory of distributions. Graduate students, professional mathematicians, and scientifically trained people with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-studyguide. Throughout, a special effort has been made to develop the theory of distributions not as an abstract edifice but rather give the reader a chance to see the rationale behind various seemingly technical definitions, as well as the opportunity to apply the newly developed tools (in the natural build-up of the theory) to concrete problems in partial differential equations and harmonic analysis, at the earliest opportunity.


The main additions to the current, second edition, pertain to fundamental solutions (through the inclusion of the Helmholtz operator, the perturbed Dirac operator, and their iterations) and the theory of Sobolev spaces (built systematically from the ground up, exploiting natural connections with the Fourier Analysis developed earlier in the monograph). 


Authors and Affiliations

  • Department of Mathematics, University of Missouri, Columbia, USA

    Dorina Mitrea

About the author

Dorina Mitrea is a Professor at University of Missouri in the Mathematics Department.

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