On non-negative contractive semigroups with non-local conditions

A second-order elliptic operator with non-local conditions in a bounded domain $Q\subset \mathbb R^n$ with boundary $\partial Q\in C^\infty$ is considered. The so-called 'non-transversal' case is investigated, that is, the case when the value of a function at each point $x\in \partial Q$ is related to the integral of this function over $\overline Q$ with respect to some Borel measure $\mu (x,dy)$. Sufficient conditions for the existence of a Feller semigroup whose infinitesimal generator is the closure in $C(\overline Q)$ of the elliptic operator under consideration are obtained.