Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems

The paper deals with a nontrivial density result of smooth functions into a Sobolev-type space involving special conditions on the boundary. Such a result is of interest when dealing with doubly elliptic problems involving two elliptic operators, one in domain and the other on the boundary of the domain. Moreover we also consider the case when a Dirichlet homogeneous boundary condition is imposed on a relatively open part of the boundary and, as a preliminary step, we shall prove an analogous result when either the entire space or the half space.