Elliptic operators with non-local Wentzell-Robin boundary conditions

TL;DR

This paper investigates elliptic differential operators with non-local boundary conditions, establishing their semigroup generation, positivity properties, and spectral characteristics on bounded Lipschitz domains.

Contribution

It introduces a comprehensive analysis of elliptic operators with non-local Wentzell-Robin boundary conditions, including semigroup generation and spectral properties.

Findings

Operators generate strongly continuous semigroups on L^2 and continuous functions.

Characterization of positivity and Markovianity of the semigroups.

Analysis of asymptotic behavior and eventual positivity of the semigroup.

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.