An operator theoretic approach to uniform (anti-)maximum principles

Abstract

Maximum principles and uniform anti-maximum principles are a ubiquitous topic in PDE theory that is closely tied to the Krein–Rutman theorem and kernel estimates for resolvents.We take up a classical idea of Takáč – to prove (anti-)maximum principles in an abstract operator theoretic framework – and combine it with recent ideas from the theory of eventually positive operator semigroups. This enables us to derive necessary and sufficient conditions for (anti-)maximum principles in a very general setting. Consequently, we are able to either prove or disprove (anti-)maximum principles for a large variety of concrete differential operators. As a bonus, for several operators that are already known to satisfy or to not satisfy anti-maximum principles, our theory gives a very clear and concise explanation of this behaviour.