Bochner formula and Bernstein type estimates on locally finite graphs

TL;DR

This paper develops Bochner and Reilly formulas for Laplacians on locally finite graphs, deriving Bernstein estimates and solutions for nonlinear heat equations, advancing analysis on discrete structures.

Contribution

It introduces new Bochner and Reilly formulas for graph Laplacians and applies them to heat equations and porous-media equations on graphs.

Findings

Derived Bernstein type estimates for heat equations on graphs

Established Reilly type formula for graph Laplacian

Obtained global positive solutions for porous-media equations

Abstract

In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly type formula of the Laplacian operator. The last one is to obtain global positive solution to porous-media equation via the use of Aronson-Benilan argument. There is not much work in the direction of the study of nonlinear heat equations on locally finite connected graphs.