ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - January 2007 - Volume 13, Issue 01  

Research Article

A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials

Nesi, Vincenzoa1 and Rogora, Enricoa1

a1 Dipartimento di Matematica, Università di Roma “La Sapienza”, Italy;  nesi@mat.uniroma1.it

Abstract

The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-r convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-r convex forms arises. In the present paper, we define the concept of extremal 2-forms  and characterize them in the rotationally invariant jointly rank-r convex case.

(Received October 27 2004)

(Revised June 30 2005)

(Online publication February 14 2007)

Key Words:

  • Compensated compactness;
  • rank-r convexity;
  • effective conductivity;
  • quadratic forms.

Mathematics Subject Classification:

  • 74Q20;
  • 49K20;
  • 35J50;
  • 74E30