Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

Abstract

We consider the first eigenvalue ${\lambda }_1(\Omega ,\sigma )$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $\Omega$ with smooth boundary, $\sigma \in \mathbf{R}$ being the Robin boundary parameter. When $\sigma >0$ we give a positive, sharp lower bound of ${\lambda }_1(\Omega ,\sigma )$ in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of $\Omega$, a lower bound of the mean curvature of $\partial \Omega$ and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean domains are obtained, which improve known estimates. Then, we extend a monotonicity result for ${\lambda }_1(\Omega ,\sigma )$ obtained in Euclidean space by Giorgi and Smits to a class of manifolds of revolution which include all space forms of constant sectional curvature. As an application, we prove that ${\lambda }_1(\Omega ,\sigma )$ is uniformly bounded below by $\frac{(n-1{)}^2}{4}$ for all bounded domains in the hyperbolic space of dimension $n$, provided that the boundary parameter $\sigma \ge \frac{n-1}{2}$ (McKean-type inequality). Asymptotics for large hyperbolic balls are also discussed