  In this paper we present an approximation result concerning the first
eigenvalue of the 1-Laplacian operator. More precisely, for $\Omega$ a bounded
regular open domain, we consider a minimisation of the functional ${\ds
\int_\Omega}|\nabla u|+n({\ds \int_\Omega} |u|-1)^2 $ over the space
$W_0^{1,1}(\Omega)$. For $n$ large enough, the infimum is achieved in some
sense on $BV(\Omega)$, and letting $n$ go to infinity this provides an
approximation of the first eigenfunction for the first eigenvalue, since the
term $n({\ds \int_\Omega} |u|^2-1)^2$ "tends" to the constraint $\|u\|_1=1$.
