Existence and regularity results for relaxed Dirichlet problems with measure data

We study the following relaxed Dirichlet problem $$\left\{ \begin{gathered} Lu + \mu u = vin\Omega , \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$ where Ω is a bounded open subset ofR ^N,Lu=?div(A?u) is an elliptic operator, μ is a positive Borel measure on Ω not charging polar sets, and v is a measure with bounded variation on Ω. We give a definition of solution for such a problem, and then prove existence and regularity results. As a consequence, the Green function for relaxed Dirichlet problems can be defined, and some of its properties are proved, including the standard representation formula for solutions.