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

TL;DR

This paper develops an abstract operator theoretic framework to establish and analyze maximum and anti-maximum principles in PDEs, unifying classical ideas with recent semigroup theory to determine these principles for various operators.

Contribution

It introduces a novel operator theoretic approach combining classical and modern techniques to characterize maximum principles in a very general setting.

Findings

Derived necessary and sufficient conditions for (anti-)maximum principles.

Proved or disproved these principles for various differential operators.

Provided clear explanations for known behaviors of certain operators regarding 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 Tak\'a\v{c} - 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.