In this paper we study the lower semicontinuous envelope with respect tothe L 1-topology of a class of isotropic functionals with lineargrowth defined on mappings from the n-dimensional ball into  ${\mathbb R}^{N}$   that are constrained to take values into a smoothsubmanifold   ${\cal Y}$   of   ${\mathbb R}^{N}$ .

New L 1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved,under a very weak dependence of the integrand with respect to the spatial variable x. Moreprecisely, only the lower semicontinuity in the sense of the 1-capacity is assumed inorder to obtain the lower semicontinuity of the functional.This condition is satisfied, for instance, by the lower approximate limit of the integrand, ifit is BV with respect to x. Under this further BV dependence,a representation formula for the relaxed functional is also obtained.

Let Ω be a smooth bounded domain in ${\bf R}^n$, n > 1, let a and f be continuous functions on $\bar\Omega$, $1^\star = {n\over n-1}$. We are concerned here with the existence of solution in $BV(\Omega)$, positive or not, to the problem:


$$ \left\{ \begin{array}{rl} -{\rm div}\ \sigma+a(x) sign\ u &= f|u|^{1^\star-2} u\cr \sigma.\nabla u &= |\nabla u|\ {\rm in}\ \Omega\cr u\ {\rm is\ not \ identically\ zero}, &-\sigma.n (u) = |u|\ {\rm on }\ \partial \Omega.\end{array}\right.$$

This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}(\Omega)$ into $L^{N\over N-1}(\Omega)$.