$p$-Laplace operator on a connected finite graph

$\def\CDp{\mathrm{CD}_p} \def\CDpp{\mathrm{CD}^\psi_p}$Let $G = (V,E)$ be a connected finite graph. In this paper we consider the $p$-Laplace operator $\Delta_p$ on $G$ for $p \in (1,+ \infty)$, which is a natural generalization of the Laplace operator. We give the definitions of the classical operators $\Delta_p, \Gamma_p, \Gamma_{2,p}$ and the $\CDp$ condition. For some $C^1$, concave function $\psi : (0,+ \infty) \to \mathbf{R}$, we define the $\psi$-operators $\Delta^\psi_p$, $\Gamma^\psi_p$, $\Gamma^\psi_{2p}$ and the $\mathrm{CD}^\psi_p$ condition. We show that the classical operators $\Delta_p$, $\Gamma_p$ and $\Gamma_{2,p}$ are directional derivatives of the corresponding $\psi$-operators. By this, we can show that the $\CDpp$ condition implies the $\CDp$ condition.We can establish the Davies’ gradient estimate for positive solutions to the $p$−Laplace parabolic equation under the $\CDpp (m,K)$ condition for some given constant $K \leq 0$. On a finite connected graph with the $\CDpp (m, 0)$ condition we can derive the logarithmic Li–Yau inequality for the $p$-Laplace parabolic equation by choosing suitable $\psi$. A similar inequality on compact manifolds with nonnegative Ricci curvature was established by Kotschwar–Ni. Based on the gradient estimate, we can derive the Harnack inequality.

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This paper was supported by the Natural of Science Foundation of Nantong City, Jiangsu Province (JC2023071).

Received 27 May 2021

Accepted 1 November 2021

Published 12 August 2024