# American Institute of Mathematical Sciences

December  2021, 20(12): 4127-4138. doi: 10.3934/cpaa.2021148

## Sharp gradient estimates on weighted manifolds with compact boundary

 1 Department of Mathematics, Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam 2 Faculty of Mathematics-Mechanics-Informatics, Hanoi University of Science (VNU), Hanoi, Vietnam 3 Thang Long Institute of Mathematics and Applied Sciences (TIMAS), Thang Long University, Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam 4 Department of Mathematics, Shanghai University, Shanghai 200444, China

* Corresponding author

Received  February 2021 Revised  August 2021 Published  December 2021 Early access  September 2021

Fund Project: Ha Tuan Dung was funded by Vingroup Joint Stock Company and supported by the Domestic PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Vingroup Big Data Institute (VINBIGDATA), code VINIF.2020.TS.12. This work also was funded by Hanoi Pedagogical University 2 Foundation for Sciences and Technology Development via grant number C.2020-SP2-07

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.

Citation: Ha Tuan Dung, Nguyen Thac Dung, Jiayong Wu. Sharp gradient estimates on weighted manifolds with compact boundary. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4127-4138. doi: 10.3934/cpaa.2021148
##### References:
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##### References:
 [1] K. Brighton, A Liouville-type theorem for smooth metric measure spaces, J. Geom. Anal., 23 (2013), 562-570.  doi: 10.1007/s12220-011-9253-5.  Google Scholar [2] R. Chen, Neumann eigenvalue estimate on a compact Riemannian manifold, Proc. Amer. Math. Soc., 108 (1990), 961-970.  doi: 10.2307/2047954.  Google Scholar [3] H. T. Dung and N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, Proc. Amer. Math. Soc., 147 (2019), 5329-5338.  doi: 10.1090/proc/14645.  Google Scholar [4] N. T. Dung and J. Y. Wu, Gradient estimates for weighted harmonic function with Dirichlet boundary condition, Nonlinear Anal., 213 (2021), Article 112498. doi: 10.1016/j.na.2021.112498.  Google Scholar [5] R. S. Hamilton, A matrix Harnack estimate for the heat equation, Commun. Anal. Geom., 1 (1993), 113-126.  doi: 10.4310/CAG.1993.v1.n1.a6.  Google Scholar [6] S. Y. Hsu, Some results for the Perelman LYH-type inequality, Discrete Contin. Dyn. Syst., 34 (2014), 3535-2554.  doi: 10.3934/dcds.2014.34.3535.  Google Scholar [7] A. V. Kolesnikov and E. Milman, Brascamp-Lieb-type inequalities on weighted Riemannian manifolds with boundary, J. Geom. Anal., 27 (2017), 1680-1702.  doi: 10.1007/s12220-016-9736-5.  Google Scholar [8] K. Kunikawa and Y. Sakurai, Yau and Souplet-Zhang type gradient estimates on Riemannian manifolds with boundary under Dirichlet boundary condition, arXiv: 2012.09374. Google Scholar [9] J. Lee, Introduction to Smooth Manifolds, Springer, New York, 2011. doi: 10.1007/978-1-4419-7940-7.  Google Scholar [10] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator,, Acta Math., 156 (1986), 153-201.  doi: 10.1007/BF02399203.  Google Scholar [11] X. D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl., 84 (2005), 1295-1361.  doi: 10.1016/j.matpur.2005.04.002.  Google Scholar [12] X. R. Olivé, Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, Proc. Amer. Math. Soc., 147 (2019), 411-426.  doi: 10.1090/proc/14213.  Google Scholar [13] Y. Sakurai, Rigidity of manifolds with boundary under a lower Ricci curvature bound, Osaka J. Math., 54 (2017), 85-119.   Google Scholar [14] Y. Sakurai, Concentration of $1$-Lipschitz functions on manifolds with boundary with Dirichlet boundary condition, arXiv: 1712.04212v4. Google Scholar [15] Y. Sakurai, Rigidity of manifolds with boundary under a lower Bakry-Émery Ricci curvature bound, Tohoku Math. J., 71 (2019), 69-109.  doi: 10.2748/tmj/1552100443.  Google Scholar [16] P. Souplet and Q. S. Zhang, Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc., 38 (2006), 1045-1053.  doi: 10.1112/S0024609306018947.  Google Scholar [17] J. P. Wang, Global heat kernel estimates, Pacific J. Math., 178 (1997), 377-398.  doi: 10.2140/pjm.1997.178.377.  Google Scholar [18] L. F. Wang, Z. Y. Zhang and Y. J. Zhou, Comparison theorems on smooth metric measure spaces with boundary, Adv. Geom., 16 (2016), 349-368.  doi: 10.1515/advgeom-2016-0022.  Google Scholar [19] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math., 28 (1975), 201-228.  doi: 10.1002/cpa.3160280203.  Google Scholar
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