Graduate Texts in Mathematics

Partial Differential Equations

Authors: Jost, Jürgen

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About this Textbook

This textbook is intended for students who wish to obtain an introduction to the theory of partial di?erential equations (PDEs, for short), in particular, those of elliptic type. Thus, it does not o?er a comprehensive overview of the whole ?eld of PDEs, but tries to lead the reader to the most important methods and central results in the case of elliptic PDEs. The guiding qu- tion is how one can ?nd a solution of such a PDE. Such a solution will, of course, depend on given constraints and, in turn, if the constraints are of the appropriate type, be uniquely determined by them. We shall pursue a number of strategies for ?nding a solution of a PDE; they can be informally characterized as follows: (0) Write down an explicit formula for the solution in terms of the given data (constraints). This may seem like the best and most natural approach, but this is possible only in rather particular and special cases. Also, such a formula may be rather complicated, so that it is not very helpful for detecting qualitative properties of a solution. Therefore, mathematical analysis has developed other, more powerful, approaches. (1) Solve a sequence of auxiliary problems that approximate the given one, and show that their solutions converge to a solution of that original pr- lem. Di?erential equations are posed in spaces of functions, and those spaces are of in?nite dimension.

Reviews

From the reviews:

MATHEMATICAL REVIEWS

"..the composition of this book is somewhat classical. Indeed the author covers the main properties of the elliptic, parabolic and hyperbolic equations. But he always adds some interesting extensions or links between these chapters…this textbook is self-contained…Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations."

"The author covers the main properties of the elliptic parabolic and hyperbolic equations. But he always adds some interesting extensions or links between these chapters. … With an appendix on general results concerning Banach or Hilbert spaces, this textbook is self-contained. … Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations." (Alain Brillard, Mathematical Reviews, 2003f)

"Jost’s book … focuses mainly on elliptic PDEs and gives a comprehensive overview of the modern theory of solving such equations. … There are extremely helpful summaries and a handful of exercises at the end of each of the 11 chapters; an appendix covers background functional analysis. … Throughout, Jost achieves an impeccable blend of motivation, orientation and analysis … . Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics." (Nick Lord, The Mathematical Gazette, Vol. 88 (512), 2004)


Table of contents (12 chapters)

  • Introduction: What Are Partial Differential Equations?

    Pages 1-6

  • The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order

    Pages 7-30

  • The Maximum Principle

    Pages 31-50

  • Existence Techniques I: Methods Based on the Maximum Principle

    Pages 51-75

  • Existence Techniques II: Parabolic Methods. The Heat Equation

    Pages 77-112

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-0-387-21595-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
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Bibliographic Information

Bibliographic Information
Book Title
Partial Differential Equations
Authors
Series Title
Graduate Texts in Mathematics
Series Volume
214
Copyright
2002
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-0-387-21595-2
DOI
10.1007/b97312
Series ISSN
0072-5285
Edition Number
1
Number of Pages
XI, 325
Topics