Highlights in the quantum theory of condensed matter *
*A symposium to honour Mario Tosi on his 72nd birthday

*edited by Fabio Beltram*

An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs

*Mariano Giaquinta, Luca Martinazzi *

This volume deals with the regularity theory for elliptic systems.

**
Highlights in the quantum theory of condensed matter
**A symposium to honour Mario Tosi on his 72nd birthday

*Edited by Fabio Beltram*

The birth of condensed matter physics in Italy is linked to a small number of very distinguished scientists. Mario Tosi, Professor of Physics of Matter at the Scuola Normale Superiore, is unquestionably among the leading figures, a true founder of the theoretical activity in the country and a true catalyst of novel research directions internationally.

This volume collects the proceedings of a symposium held at Scuola Normale Superiore di Pisa, designed to show Mario Tosi’s broad, deep influence in very diverse areas of the quantum theory of condensed matter.

The topics covered in the volume representing the breadth of his interests and the highlights in the quantum theory of condensed matter:

- Liquids
- Electronic states in complex structures
- Quantum degenerate gases
- Many-body physics

**Argomenti correlati **

*10-11 settembre 2004
*

**Highlights in the quantum theory of condensed matter**

**A symposium to honour Mario Tosi on his 72nd birthday**

Scuola Normale Superiore, NEST-INFM

An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs

*Mariano Giaquinta, Luca Martinazzi
*

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 on the occasion of 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 is in 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; finally, harmonic maps and minimal graphs in codimension 1 and greater than 1.

**Edizioni della Normale,
**Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56100 Pisa,

Tel: (0039) 050 509220, Fax: (0039) 050 509278

email: edizioni@sns.it

Distribuzione:

per l’Italia RCSLibri , per l’estero Casalini Libri