Phragm\`en-Lindel\"of type theorems for elliptic equations on infinite graphs

TL;DR

This paper extends Phragmèn-Lindelöf theorems to elliptic equations on infinite graphs, establishing uniqueness of solutions under growth conditions and analyzing the influence of graph geometry and potential decay.

Contribution

It introduces conditions under which Phragmèn-Lindelöf principles hold for elliptic equations on infinite graphs, including bounds on outer degree and potential decay.

Findings

Phragmèn-Lindelöf principles are valid under bounded outer degree.

Uniqueness of solutions is guaranteed with suitable growth constraints.

Conditions on potential decay are shown to be optimal in certain graph classes.

Abstract

We investigate the validity of the Phragm\`en-Lindel\"of principle for a class of elliptic equations with a potential, posed on infinite graphs. Consequently, we get uniqueness, in the class of solutions satisfying a suitable growth condition at infinity. We suppose that the {\it outer degree (or outer curvature)} of the graph is bounded from above, and we allow the potential to go to zero at infinity in a controlled way. Finally, we discuss the optimality of the conditions on the potential and on the outer degree on special graphs.