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

Xiaolong Li; Kui Wang

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arXiv · 2002.06472

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

Xiaolong Li·Kui Wang

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Abstract

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

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Xiaolong Li, Kui Wang. 2020-02-15. First Robin Eigenvalue of the $p$-Laplacian on Riemannian Manifolds. https://arxiv.org/abs/2002.06472

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