Existence and comparison results for an elliptic equation involving the $1$-Laplacian and $L^1$-data

Abstract

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.