Relaxation in BV of Integral Functionals Defined on Sobolev Functions with Values in the Unit Sphere

We study the relaxation with respect to the $L^1$ norm of integral functionals of the type $$F(u)=\int_\Omega f(x,u,\nabla u)\,dx\,\quad u\in W^{1,1}(\Omega;S^{d-1})$$ where $\Omega$ is a bounded open set of $R^N$, $S^{d-1}$ denotes the unite sphere in $R^d$, $N$ and $d$ being any positive integers, and $f$ satisfies linear growth conditions in the gradient variable. In analogy with the unconstrained case, we show that, if, in addition, $f$ is quasiconvex in the gradient variable and satisfies some technical continuity hypotheses, then the relaxed functional $\overline F$ has an integral representation on $BV(\Omega;S^{d-1})$ of the type $$\bar F(u)=\int_{\Omega}f(x,u,\nabla u)\,dx+\int_{S(u)}K(x,u^-,u^+,\nu_u)\,d{\cal H}^{N-1} + \int_\Omega f^\infty (x,u,d C(u)),$$ where the suface energy density $K$ is defined by a suitable Dirichlet-type problem.