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Bull. Korean Math. Soc. 2023; 60(2): 325-338

Online first article January 26, 2023      Printed March 31, 2023

https://doi.org/10.4134/BKMS.b220120

Copyright © The Korean Mathematical Society.

Some integral inequalities for the Laplacian with density on weighted manifolds with boundary

Fanqi Zeng

Xinyang Normal University

Abstract

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In this paper, we derive a Reilly-type inequality for the Laplacian with density on weighted manifolds with boundary. As its applications, we obtain some new Poincar\'{e}-type inequalities not only on weighted manifolds, but more interestingly, also on their boundary. Furthermore, some mean-curvature type inequalities on the boundary are also given.

Keywords: Reilly-type inequality, generalized Ricci curvature, generalized mean curvature

MSC numbers: 53C21, 58J32

Supported by: This work was supported by NSFC (No.11971415), Natural Science Foundation of Henan (No.212300410235), and the Key Scientific Research Program in Universities of Henan Province (Nos.21A110021, 22A110021) and Nanhu Scholars Program for Young Scholars of XYNU (No.2019).

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