Existence and regularity results for relaxed Dirichlet problems with measure data.

We study the following relaxed Dirichlet problem Lu + mu u = nu in Omega, u = 0 on partial derivative Omega, where Omega is a bounded open, subset of R-N, Lu = -div(A del u) is an elliptic operator, mu is a positive Borel measure on Omega not charging polar sets, and nu is a measure with bounded variation on Omega. 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.