RELAXATION OF QUASI-CONVEX FUNCTIONALS IN BV (OMEGA, R(P)) FOR INTEGRANDS F(X, U, DEL U)

In this paper it is shown that if f(x, u, .) is a quasiconvex function with linear growth, then the relaxed functional in BV(OMEGA, R(p)) of u --> integral/OMEGA f(x, u(x), delu(x)) dx with respect to the L1 topology has an integral representation of the form F(u) = integral/OMEGA f(x, u(x), delu(x)) dx + integral/SIGMA(u) K(x, u-(x), u+(x), v(x)) dH(N-1)(x) + integral/OMEGA f(infinity)(x, u(x), dC(u)) where Du = delu dx + (u+ - u-) x v dH(N-1) L SIGMA(u) + C(u). The proof relies on a blow-up argument introduced by FONSECA & MULLER in the case where u is-an-element-of W1,1 and on a recent result by ALBERTI showing that the Cantor part C(u) is rank-one valued.