Li-Yau inequality and related properties on metric star graphs

TL;DR

We address gradient estimates for the heat equation on a metric star graph $\mathcal{G}$ with Kirchhoff transmission at the central vertex. By exploiting an explicit heat-kernel representation $\Gamma_i$ on $\mathcal{G}$, we derive a Li-Yau type gradient bound for positive solutions $u=P_t\phi$, namely $\partial_{xx}(\ln(u))\ge -\frac{1}{2t}-(1-2\alpha_i)\mathcal{I}_i(x,t)$ with a positive, vertex-dependent term $\mathcal{I}_i(x,t)$ that vanishes at the vertex and reflects graph asymmetry. The paper further obtains a Harnack inequality for the heat equation and a Liouville theorem stating that every bounded harmonic function on $\mathcal{G}$ is constant, highlighting the sharpness relative to the Euclidean case through examples. It also discusses the interpretation of the extra term and outlines open questions about its relation to initial data and $\alpha_i$.

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 $\mG$.

Figures (1)

• Figure 1: A star graph with $5$ edges.

Theorems & Definitions (10)

• Remark 1.1
• Remark 1.2
• Theorem 2.1
• proof
• Remark 2.2
• Remark 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof