An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs / Publications of the Scuola Normale Superiore (PDF)

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 thiscontext, 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.