Relaxation of singular functionals defined on sobolev spaces

In this paper, we consider a Borel measurable function on the space of m×n matrices f:Mm×n → taking the value +∞, such that its rank-one-convex envelope Rf is finite and satisfies for some fixed p<1: −c0≤Rf(F)c≤(1+‖F‖p) for all FϵMm×n, where c, c0>0. Let be a given regular bounded open domain of Rn. We define on W1, p(Ω, Rm) the functional I(u)= ∫Ωf(∇u(x)) dx. Then, under some technical restrictions on f, we show that the relaxed functional r for the weak topology of W1, p(Ω, Rm) has the integral representation: I(u)=∫Ω Q[Rf](∇u(x)) dx; where for a given function g, Qg denotes its quasiconvex envelope. © 2000 EDP Sciences, SMAI.