Inequalities for Eigenvalues of a System of Equations of Elliptic Operator in Weighted Divergence Form on Metric Measure Space

Abstract

Let A be a symmetric and positive definite (1, 1) tensor on a bounded domain Ω in an n-dimensional metric measure space (ℝⁿ, 〈,〉, e^−ϕdv). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form

$$\left\{ {\begin{array}{*{20}{l}} {{\mathbb{L}_{_{A,\varphi }}}\text{u} + \alpha [\nabla (\text{div}\text{u}) - \nabla {\varphi }\text{div}\text{u}] = -\varsigma \text{u},\;\;in\;\;\Omega ,} \\ {u{|_{\partial \Omega }} = 0,} \end{array}} \right.$$

where ${L_{A,\varphi }} = {\rm{div}}\left( {A\nabla \left( \cdot \right)} \right) - \left\langle {A\nabla \varphi ,\nabla \left( \cdot \right)} \right\rangle$, α is a nonnegative constant and u is a vector-valued function. Some universal inequalities for eigenvalues of this problem are established. Moreover, as applications of these results, we give some estimates for the upper bound of ςk+1 and the gap of ςk+1 −ςk in terms of the first k eigenvalues. Our results contain some results for the Lame system and a system of equations of the drifting Laplacian.