An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs
Authors: Giaquinta, Mariano, Martinazzi, Luca
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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^ptheory both with and without potential theory, including the CalderonZygmund theorem, Harnack's and De GiorgiMoserNash 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 2dimensional 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 CoifmanLionsMeyerSemmes.
 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
Pages 116

Direct methods
Pages 1735

Hilbert space methods
Pages 3759

L2regularity: The Caccioppoli inequality
Pages 6173

Schauder estimates
Pages 7595

Table of contents (11 chapters)
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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

 Mariano Giaquinta
 Luca Martinazzi
 Series Title
 Lecture Notes (Scuola Normale Superiore)
 Series Volume
 11
 Copyright
 2012
 Publisher
 Edizioni della Normale
 Copyright Holder
 Edizioni della Normale
 eBook ISBN
 9788876424434
 DOI
 10.1007/9788876424434
 Softcover ISBN
 9788876424427
 Edition Number
 2
 Number of Pages
 XIII, 370
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