*C.N.R.- Gruppo Nazionale Analisi Matematica Probabilità
e Applicazioni*

School on

*HOMOGENIZATION TECHNIQUES
AND ASYMPTOTIC METHODS FOR PROBLEMS WITH MULTIPLE SCALES*

*Part I - 17-21 September 2001*

Dipartimento di Matematica - Politecnico di Torino, Turin

*Part II - December 3-5, 2001*

Istituto per le Applicazioni del Calcolo, Rome

**Lecturers and Courses**:

**TURIN**

A. Braides (Università di Roma `Tor Vergata'): From discrete to continuous variational problems: an introduction

A. Defranceschi (Università di Parma): Relaxation
problems for bulk and interfacial energies

**ROME**

G. Francfort (Universite' Paris Nord): H-measures and semi-classical measures: an introduction

R. Peirone (Universita` di Roma `Tor Vergata'): An
introduction to Dirichlet Forms on fractals

The School in Turin had to be shortened due to the September 11 events. The following courses had to be cancelled:

G. Chechkin (University of Moscow): Homogenization in perforated domains

A. Piatnitski (University of Moscow): Random
homogenization: a basic introduction

**Aim of the School**

The School is part of a project of the G.N.A.M.P.A. whose aim is to analyze techniques developed from various groups of research (operating in particular in Russia, France and Italy) relative to the interaction of phenomena of homogenization with other limit procedures as dimension reduction (mathematical theories of plates and thin films), the theory of phase transitions, processes of discretization with applications both numerical and theoretical (limits of finite-difference schemes), limits of perforated domains (relaxed Dirichlet problems), limits of sets with strongly-oscillating boundaries. A great variety of instruments has been developed in order to study such problems: from G-convergence, developed both in Italy and France and in Russia, to the Gamma-convergence of De Giorgi, the compensated compactness of Murat and Tartar, from the use of Young measures, to two-scale convergence, H-convergence, etc. These methods have often remained confined to specialistic areas, but interesting result have been recently obtained by combining these techniques, in problems where more scales are present, typically problems in which the processes hinted at above interact. In many cases, in the `superpositions' of more problems new phenomena are described that are not characteristic of the single problems. As an example, homogenization methods have allowed to characterize interesting phenomena of oscillations in problems of non-convex thin films, the application of capacitary methods to the homogenization of degenerate materials have emphasized non-local phenomena, the methods of Gamma-convergence combined to those of Young measures have emphasized phenomena with more scales in non convex homogenous materials, etc. In this School we have given an introduction to some of the techniques developed to study oscillatory phenomena with multiple scales.

The School was mainly thought for Ph D students or researchers
in the area of (applied) Analysis, but no particular background was required.

**Co-ordinator of the Project:**

Valeria Chiadò Piat,
Dipartimento di Matematica

Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129
Torino

e-mail: vchiado@polito.it

**Sponsors**: the Turin section of the School has been
co-sponsored by 'cofin 2000, Gruppo Nazionale Calcolo delle Variazioni,
unita` locali del Politecnico di Milano (Dipartimento
di Matematica Francesco Brioschi) e dell'Università di Parma'
and SISSA, Trieste

In collaboration with: TMR Project *Homogenization
and Multiple Scales*