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Birkhäuser

An Invitation to Variational Methods in Differential Equations

  • Textbook
  • © 2007

Overview

Authors:
  1. David G. Costa

    1. Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, USA

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  • Serves as a sampling of topics in critical point theory
  • Ideal for graduate students and researchers interested in differential equations and nonlinear analysis
  • Applications immediately follow each result for easy assimilation by the reader

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • David G. Costa
    Pages 1-5

  3. Critical Points Via Minimization

    • David G. Costa
    Pages 7-18

  4. The Deformation Theorem

    • David G. Costa
    Pages 19-27

  5. The Mountain-Pass Theorem

    • David G. Costa
    Pages 29-38

  6. The Saddle-Point Theorem

    • David G. Costa
    Pages 39-47

  7. Critical Points under Constraints

    • David G. Costa
    Pages 49-62

  8. A Duality Principle

    • David G. Costa
    Pages 63-73

  9. Critical Points under Symmetries

    • David G. Costa
    Pages 75-85

  10. Problems with an S1-Symmetry

    • David G. Costa
    Pages 87-97

  11. Problems with Lack of Compactness

    • David G. Costa
    Pages 99-113

  12. Lack of Compactness for Bounded Ω

    • David G. Costa
    Pages 115-124

  13. Appendix

    • David G. Costa
    Pages 125-129

  14. Back Matter

    Pages 131-138
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Keywords

About this book

This little book is a revised and expanded version of one I wrote for the "VIII Latin American School of Mathematics" [29], in Portuguese, based on which I have periodically taught a topics course over the last 18 years. As the - graduate students tle suggests, it is an introductory text. It is addressed to of mathematics in the area of differential equations/nonlinear analysis and to mathematicians in other areas who would like to have a first exposure to so called variational methods and their applications to PDEs and ODEs. Afterwards, the reader can choose from some excellent and more compreh- sive texts, which already exist in the literature but require somewhat more maturity in the area. We present a cross section of the area of variational methods, with a m- imum (no "pun" intended) of material, but clearly illustrating through one or two examples each of the results that we have chosen to present. So, besides the first motivating chapter and an appendix, there are only ten short chapters (with three or, at most, four sections each) through which the reader is quickly exposed to a few basic aspects of the beautiful area of variational methods and applications to differential equations. In fact, the reader may initially skip some of the more technical proofs of the main theorems, concentrating instead on the applications that are given.

Reviews

From the reviews:

"This book is intended to be an introduction to variational methods for ODEs and PDEs. … Overall, this is a well-written book on the variational methods … . The author clearly loves and knows the subject area and it is very good at providing an overview to the area … . As such, a good guided reading course for graduate students could be made from this book, covering one chapter per session (or two)." (David A. W. Barton, Dynamical Systems Magazine, April, 2008)

"This little book consists of a very clear introduction to variational methods and their applications to ODEs and PDEs. … the bibliography contains basic items on the subject and also turns out to be very useful for further reading. In my opinion the book should be strongly recommended to anyone—graduate student or researcher—who is interested in variational methods and their applications to differential equations." (Salvatore A. Marano, Mathematical Reviews, Issue 2008 k)

“The intention of the author is to provide a first introduction to variational methods in solving ODEs and PDEs for graduate students. The book is a concise collection of some fundamental aspects of this area … . It presents a nice and direct approach to these central topics and is written with great care and clarity. It can be warmly recommended to anyone wishing to enter this active area of research.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)

Authors and Affiliations

  • Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, USA

    David G. Costa

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