Gradient estimates for weighted harmonic function with Dirichlet boundary condition

TL;DR

This paper establishes gradient estimates for positive weighted harmonic functions with Dirichlet boundary conditions on smooth metric measure spaces, leading to Liouville type results without assumptions on the potential function.

Contribution

It extends Yau's gradient estimate to weighted harmonic functions with boundary conditions on metric measure spaces, independent of the potential function.

Findings

Gradient estimates for positive $f$-harmonic functions with Dirichlet boundary conditions.

Liouville type theorem for bounded $f$-harmonic functions.

Results hold without assumptions on the potential function $f$.

Abstract

We prove a Yau's type gradient estimate for positive $f$-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry-Emery Ricci tensor and the weighted mean curvature are bounded below. As an application, we give a Liouville type result for bounded $f$-harmonic functions with the Dirichlet boundary condition. Our results do not depend on any assumption on the potential function $f$.