Mathematical Research Letters

Volume 28 (2021)

Number 5

Sharp lower bound for the first eigenvalue of the weighted $p$-Laplacian, II

Pages: 1459 – 1479



Xiaolong Li (Department of Mathematics, University of California, Irvine, Calif., U.S.A.; and Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, Kansas, U.S.A.)

Kui Wang (School of Mathematical Sciences, Soochow University, Suzhou, China)


Combined with our previous work [14], we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1 \lt p \lt \infty$ on a compact Bakry–Émery manifold $(M^n, g, f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\operatorname{Ric}+ \nabla^2 f \geq \kappa g$ for some $\kappa \in \mathbb{R}$.

The research of the second author is supported by NSFC No.11601359.

Received 18 December 2019

Accepted 3 May 2020

Published 16 August 2022