First Robin eigenvalue of the p-Laplacian on Riemannian manifolds

Abstract

We consider the first Robin eigenvalue $$\lambda _p(M,\alpha )$$ for the p-Laplacian on a compact Riemannian manifold M with nonempty smooth boundary, with $$\alpha \in \mathbb {R}$$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $$\lambda _p(M,\alpha )$$ . Secondly, when $$\alpha >0$$ we establish sharp lower bound of $$\lambda _p(M,\alpha )$$ 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 $$\alpha <0$$ . Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the p-Laplacian when letting $$\alpha \rightarrow +\infty $$ .