A characterization of the individual maximum and anti-maximum principle

TL;DR

This paper investigates the conditions under which individual maximum and anti-maximum principles hold for differential operators, revealing that these principles are equivalent to a domination condition involving the leading eigenfunction.

Contribution

It establishes that the domination condition is necessary and sufficient for the simultaneous validity of both principles, clarifying their applicability to differential operators.

Findings

The domination condition characterizes when both principles hold.

Many concrete differential operators do not satisfy the anti-maximum principle.

The study links spectral properties to maximum principles in differential operators.

Abstract

Abstract approaches to maximum and anti-maximum principles for differential operators typically rely on the condition that all vectors in the domain of the operator are dominated by the leading eigenfunction of the operator. We study the necessity of this condition. In particular, we show that under a number of natural assumptions, so-called individual versions of both the maximum and the anti-maximum principle simultaneously hold if and only if the aforementioned domination condition is satisfied. Consequently, we are able to show that a variety of concrete differential operators do not satisfy an anti-maximum principle.