Univalent σ-harmonic mappings: applications to composites

¹ Dipartimento di Scienze Matematiche, Universitá degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy; alessang@univ.trieste.it.
² Dipartimento di Matematica, Universitá di Roma, La Sapienza, P. le A. Moro 2, 00185 Rome, Italy; nesi@mat.uniroma1.it.

Received: 15 June 2001
Revised: 23 January 2002

Abstract

This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all the two dimensional G-closure problems in conductivity.