Sub-supersolution method for a quasilinear elliptic problem involving the 1-laplacian operator and a gradient term

Abstract

In this work we study a quasilinear elliptic problem involving the 1-laplacian operator and a gradient term. The problem requires the definition of a suitable sense of solution, which allows us to show the existence of a solution in BV(Ω), having no jump part. Despite the lack of regularity of the solutions, we develop a sub-supersolution approach, together with a thorough analysis of the distributional derivative of the functions in BV(Ω).