Boundary value problems with measures for elliptic equations with singular potentials

We study the boundary value problem with Radon measures for nonnegative solutions of LVu:=−Δu+Vu=0 in a bounded smooth domain Ω, when V is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure μ on ∂Ω so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions. In Appendix A A. Ancona solves a question raised by M. Marcus and L. Véron concerning the vanishing set of the Poisson kernel of LV for an important class of potentials V.