Talk by Sahiba Arora

On July 7th, 2021, Sahiba Arora (TU Dresden) gave a talk about "Uniform maximum and anti-maximum principles" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.


Show Abstract

Extensive literature has been devoted to study the operators for which the maximum and/or the anti-maximum principle holds. Combining an idea of Takáč (1996) with those from the recent theory of eventually positive $C_0$-semigroups, we look at some necessary and sufficient conditions for (anti-)maximum principles to hold in an abstract setting of Banach lattices.

More precisely, if $A : \operatorname{dom}(A) \subseteq E\to E$ is a closed, densely defined, and real operator on a complex Banach lattice $E$ (or in particular, an $L^p$-space), then we consider the equation
(\lambda-A)u =f
for real numbers $\lambda$ in the resolvent set of $A$. We ask whether $f\geq 0$ implies $u\geq 0$ for $\lambda$ in a right neighbourhood of an eigenvalue. In this case, we say that the maximum principle is satisfied. Analogously, when the implication $f\geq 0$ implies $u\leq 0$ holds for $\lambda$ in a left neighbourhood of an eigenvalue, we say that the anti-maximum principle holds.

We will also see how these abstract results can be applied to various concrete differential operators and illustrate how several previously known results about (anti-)maximum principles can be proved via this theory. This is joint work with Jochen Glück.

Video of the talk

Liza Schonlau | 10.05.2024