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

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 Takac - 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This article investigates the maximum principles and uniform anti-maximum principles in PDE theory, combining classical ideas from abstract operator theory with recent ideas from the theory of eventually positive operator semigroups. The necessary and sufficient conditions for (anti-)maximum principles are derived in a very general setting, allowing for the proof or disproof of (anti-)maximum principles for various concrete differential operators. Additionally, the theory provides a clear and concise explanation for the behavior of operators that already satisfy or do not satisfy anti-maximum principles.