Boundary value problems and variational inequalities, associated with second order elliptic operators, will be studied in a Hilbert space framework. In this space, functions will have (at least) locally square integrable derivatives of order up to two. Also the conormal derivative, extended by continuity, will be square integrable on the boundary of the region considered. Criteria for approximating elements of the Hilbert space by smooth functions will be given and thus closed convex sets, associated with inequalities on the boundary, exist.