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

In this paper, we investigate the first non-zero eigenvalue problem of the following operator divA∇f=0inΩ,∂f∂v|=pfon∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} \mathrm {div} A\nabla {f}\mathrm =0 \quad \hbox {in}\quad \Omega ,\\ \frac{\partial f}{\partial v}|=pf\ \quad \hbox {on}\quad \partial \Omega ,\\ \end{array} \right. \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a compact bounded domain in an m-dimensional complete Riemannian manifold Mm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^{m}$$\end{document}, v is the outward unit normal vector field of ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document} and A is a positive definite symmetric (1,1)-tensor on Mm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^{m}$$\end{document}. 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 ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}.