Li-Yau inequality and related properties on metric star graphs

TL;DR

This paper establishes a Li-Yau gradient estimate for heat equations on metric star graphs, leading to Harnack inequalities and Liouville properties, by leveraging an explicit heat kernel representation.

Contribution

It introduces a novel Li-Yau inequality for metric star graphs using explicit heat kernel formulas, extending classical results to this geometric setting.

Findings

Proves a Li-Yau gradient estimate for heat equations on star graphs

Derives Harnack inequalities from the gradient estimate

Establishes Liouville property for bounded harmonic functions

Abstract

We prove a Li-Yau gradient estimate for positive solutions to the heat equation defined on a metric star graph $\mG$ given by the heat kernel formula. As consequence, we derive a Harnack estimate and a Liouville property for bounded harmonic functions. The argument exploits an explicit representation formula for the heat kernel on $\mG$.