First Robin Eigenvalue of the $p$-Laplacian on Riemannian Manifolds

Xiaolong Li, Kui Wang

Published 2020-02-15Version 1

We consider the first Robin eigenvalue $\lambda_p(M,\alpha)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty boundary, with $\alpha \in \mathbb{R}$ being the Robin parameter. We prove eigenvalue comparison theorems of Cheng type for $\lambda_p(M,\alpha)$. For $\alpha>0$, we establish sharp lower bound estimates of $\lambda_p(M,\alpha)$ in terms of the dimension, inradius, Ricci lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. For $\alpha<0$, the lower bound becomes an upper bound. Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the $p$-Laplacian when letting $\alpha \to +\infty$.