An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs

  • Mariano Giaquinta
  • Luca Martinazzi

Part of the Publications of the Scuola Normale Superiore book series (PSNS)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Mariano Giaquinta, Luca Martinazzi
    Pages 1-16
  3. Mariano Giaquinta, Luca Martinazzi
    Pages 17-35
  4. Mariano Giaquinta, Luca Martinazzi
    Pages 37-59
  5. Mariano Giaquinta, Luca Martinazzi
    Pages 61-73
  6. Mariano Giaquinta, Luca Martinazzi
    Pages 75-95
  7. Mariano Giaquinta, Luca Martinazzi
    Pages 97-135
  8. Mariano Giaquinta, Luca Martinazzi
    Pages 137-166
  9. Mariano Giaquinta, Luca Martinazzi
    Pages 167-203
  10. Mariano Giaquinta, Luca Martinazzi
    Pages 205-228
  11. Mariano Giaquinta, Luca Martinazzi
    Pages 229-291
  12. Mariano Giaquinta, Luca Martinazzi
    Pages 293-354
  13. Back Matter
    Pages 355-369

About this book


This volume deals with the regularity theory for elliptic systems. We may find the origin of such a theory in two of the problems posed by David Hilbert in his celebrated lecture delivered during the International Congress of Mathematicians in 1900 in Paris:

19th problem: Are the solutions to regular problems in the Calculus of Variations always necessarily analytic?

20th problem: does any variational problem have a solution, provided that certain assumptions regarding the given boundary conditions are satisfied, and provided that the notion of a solution is suitably extended?

During the last century these two problems have generated a great deal of work, usually referred to as regularity theory, which makes this topic quite relevant in many fields and still very active for research.
However, the purpose of this volume, addressed mainly to students, is much more limited. We aim to illustrate only some of the basic ideas and techniques introduced in this context, confining ourselves to important but simple situations and refraining from completeness. In fact some relevant topics are omitted.
Topics include: harmonic functions, direct methods, Hilbert space methods and Sobolev spaces, energy estimates, Schauder and Lp-theory both with and without potential theory, including the Calderon-Zygmund theorem, Harnack's and De Giorgi-Moser-Nash theorems in the scalar case and partial regularity theorems in the vector valued case; energy minimizing harmonic maps and minimal graphs in codimension 1 and greater than 1.

In this second deeply revised edition we also included the regularity of 2-dimensional weakly harmonic maps, the partial regularity of stationary harmonic maps, and their connections with the case p=1 of the Lp theory, including the celebrated results of Wente and of Coifman-Lions-Meyer-Semmes.


elliptic systems harmonic maps minimal graphs partial differential equations regularity theory

Authors and affiliations

  • Mariano Giaquinta
    • 1
  • Luca Martinazzi
    • 2
  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Rutgers UniversityPiscatawayUSA

Bibliographic information

  • DOI
  • Copyright Information Edizioni della Normale 2012
  • Publisher Name Edizioni della Normale, Pisa
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-88-7642-442-7
  • Online ISBN 978-88-7642-443-4
  • About this book