Gradient estimate for the Poisson equation and the non-homogeneous heat equation on compact Riemannian manifolds

TL;DR

This paper extends gradient estimates for positive solutions of the Poisson and non-homogeneous heat equations on compact Riemannian manifolds, broadening the understanding of these equations beyond harmonic functions.

Contribution

It provides new gradient estimates for positive solutions to the Poisson and heat equations on compact Riemannian manifolds, extending existing results for harmonic functions.

Findings

Extended gradient estimates to non-harmonic solutions

Applicable to compact Riemannian manifolds

Generalized previous harmonic function results

Abstract

In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.