On some nonlinear partial differential equations involving the 1-Laplacian

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 $\|u\|_1=1$.