Elliptic operators with non-local Wentzell-Robin boundary conditions

Abstract

In this article, we study strictly elliptic, second-order differential operators on a bounded Lipschitz domain in $\mathbb{R}^d$, subject to certain non-local Wentzell-Robin boundary conditions. We prove that such operators generate strongly continuous semigroups on $L^2$-spaces and on spaces of continuous functions. We also provide a characterisation of positivity and (sub-)Markovianity of these semigroups. Moreover, based on spectral analysis of these operators, we discuss further properties of the semigroup such as asymptotic behaviour and, in the case of a non-positive semigroup, the weaker notion of eventual positivity of the semigroup.