A Reilly Inequality for the First Non-zero Eigenvalue of a Class of Operators on Riemannian Manifold

Abstract

In this paper, we investigate the first non-zero eigenvalue problem of the following operator $$\begin{aligned} \left\{ \begin{array}{l} \mathrm {div} A\nabla {f}\mathrm =0 \quad \hbox {in}\quad \Omega ,\\ \frac{\partial f}{\partial v}|=pf\ \quad \hbox {on}\quad \partial \Omega ,\\ \end{array} \right. \end{aligned}$$ where \(\Omega \) is a compact bounded domain in an m-dimensional complete Riemannian manifold \(M^{m}\), v is the outward unit normal vector field of \(\partial \Omega \) and A is a positive definite symmetric (1,1)-tensor on \(M^{m}\). By the Rayleigh-Ritz inequality and Hsiung–Minkowski formulas, we derive an upper bound for the first non-zero eigenvalue of these operators on bounded domain of complete manifolds isometrically immersed in a Euclidean space or a unit Sphere in terms of the r-th mean curvatures of its boundary \(\partial \Omega \).