Gradient estimates for weighted harmonic function with Dirichlet boundary condition

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.