  This paper is devoted to analyse the Dirichlet problem for a nonlinear
elliptic equation involving the $1$--Laplacian and a total variation term, that
is, the inhomogeneous case of the equation arising in the level set formulation
of the inverse mean curvature flow. We study this problem in an open bounded
set with Lipschitz boundary.
  We prove an existence result and a comparison principle for non--negative
$L^1$--data. Moreover, we search the summability that the solution reaches when
more regular $L^p$--data, with $1<p<N$, are considered and we give evidence
that this summability is optimal.
  To prove these results, we apply the theory of
$L^\infty$--divergence--measure fields which goes back to Anzellotti (1983).
The main difficulties of the proofs come from the absence of a definition for
the pairing of a general $L^\infty$--divergence--measure field and the gradient
of an unbounded $BV$--function.
