Gradient estimates of nonlinear equation on complete noncompact metric measure space with compact boundary

TL;DR

This paper derives gradient estimates for positive solutions of nonlinear equations on complete noncompact metric measure spaces with boundary, extending Li-Yau and Hamilton's estimates to both parabolic and elliptic cases.

Contribution

It provides new gradient estimates for nonlinear equations on metric measure spaces with boundary, including Li-Yau and Hamilton types, for both parabolic and elliptic equations.

Findings

Established Li-Yau type gradient estimates for parabolic equations.

Derived Hamilton's type gradient estimates for the same class of equations.

Obtained gradient bounds for elliptic equations on noncompact metric measure spaces.

Abstract

In this paper, firstly, we study gradient estimates for positive solution of the following equation \begin{equation*} \Delta_\xi(u)-\partial_t u- q u =A(u),t\in (-\infty,\infty) \end{equation*} on metric measure space $ (M,g,e^{-\xi}\mathrm{d} v_g)$ with boundary , where $ \Delta_\xi=\Delta +\left \langle \nabla\cdot , \nabla \xi\right \rangle $. For this equation, we derive Li-Yau type gradient estimates and Hamilton's type gradient estimates. Secondly, we obtain gradient estimates for positive solution of the following elliptical type equation \begin{equation} \Delta_\xi(u)- q u =A(u)\end{equation} on complete noncompact metric measure space $(M,g,e^{-\xi}\mathrm{d} v_g)$ with boundary.