Lecture Notes (Scuola Normale Superiore)

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

Authors: Giaquinta, Mariano, Martinazzi, Luca

  • Covers both classical and recent topics
  • Very few prerequisites
  • Excellent introduction to the subject
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About this Textbook

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 L^p-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 L^p theory, including the celebrated results of Wente and of Coifman-Lions-Meyer-Semmes.

About the authors

Mariano Giaquinta is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equation. He is currently professor of mathematics at the Scuola Normale Superiore di Pisa [1] [2] and he is the director of De Giorgi center at Pisa. Luca Martinazzi is professor of mathematics at the University of Basel, Switzerland.

Table of contents (11 chapters)

  • Harmonic functions

    Giaquinta, Mariano (et al.)

    Pages 1-16

  • Direct methods

    Giaquinta, Mariano (et al.)

    Pages 17-35

  • Hilbert space methods

    Giaquinta, Mariano (et al.)

    Pages 37-59

  • L2-regularity: The Caccioppoli inequality

    Giaquinta, Mariano (et al.)

    Pages 61-73

  • Schauder estimates

    Giaquinta, Mariano (et al.)

    Pages 75-95

Buy this book

eBook $19.99
price for USA (gross)
  • ISBN 978-88-7642-443-4
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $29.95
price for USA
  • ISBN 978-88-7642-442-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs
Authors
Series Title
Lecture Notes (Scuola Normale Superiore)
Series Volume
11
Copyright
2012
Publisher
Edizioni della Normale
Copyright Holder
Edizioni della Normale
eBook ISBN
978-88-7642-443-4
DOI
10.1007/978-88-7642-443-4
Softcover ISBN
978-88-7642-442-7
Edition Number
2
Number of Pages
XIII, 370
Topics