Relaxation of Quasilinear Elliptic Systems via A-quasiconvex Envelopes

Institute of Mathematics and Computer Science, University of Latvia, 1459 Riga, Latvia; uldis.raitums@mii.lu.lv.

Received: May 2001
Received: November 2001

Abstract

We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems where Ω ⊂ Rⁿ is a bounded Lipschitz domain, F_s are strictly convex smooth functions with quadratic growth and . We show that WZ is the zero level set for an integral functional with the integrand being the A-quasiconvex envelope for a certain function and the operator A = (curl,div)^m. If the functions F_s are isotropic, then on the characteristic cone Λ (defined by the operator A) coincides with the A-polyconvex envelope of and can be computed by means of rank-one laminates.