Gradient estimates of nonlinear equation on complete noncompact metric measure space with compact boundary
| Autor: | Cao, Xiangzhi |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: The main result of this paper, we think, is not worth publishing, since the condition is too strong and not natural. This paper is needed to improved in the future |
| Databáze: | arXiv |
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