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

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

Received: 27 October 2004
Revised: 30 June 2005

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.