Sharp gradient estimates on weighted manifolds with compact boundary

TL;DR

This paper establishes precise gradient bounds for positive solutions to the weighted heat equation on manifolds with boundary, leading to Liouville theorems and growth restrictions, refining recent related results.

Contribution

It provides sharp gradient estimates on weighted manifolds with boundary and derives Liouville theorems, advancing the understanding of heat equations in this setting.

Findings

Sharp gradient estimates for positive solutions

Liouville theorems for ancient solutions with boundary conditions

Refinement of previous results by Kunikawa and Sakurai

Abstract

In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.