Relaxation of singular functionals defined on Sobolev spaces

In this paper, we consider a Borel measurable function on the space of m x n matrices f:M-m x n --> (R) over bar taking the value +infinity, such that its rank-one-convex envelope Rf is finite and satisfies for some fixed p>1: -c(0) less than or equal to Rf(F)less than or equal toc(1+ parallel toF parallel to (p)) for all F is an element of M-m x n, where c,c(0) >0. Let Omega be a given regular bounded open domain of R-n. We define on W-1,W-p (Omega; R-m) the functional I(u)= f(Omega)f(delu(x)) dx. Then, under some technical restrictions on f, we show that the relaxed functional (I) over bar for the weak topology of W-1,W-p(Omega; R-m) has the integral representation: (I) over bar (u)=f(Omega)Q[Rf](delu(x)) dx, where for a given function g, Qg denotes its quasiconvex envelope.