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.

中文翻译：

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度量空间上加权发散形式的椭圆算子方程组特征值的不等式

设 A 是 n 维度量空间 (ℝn, <,>, e−ϕdv) 中的有界域 Ω 上的对称正定 (1, 1) 张量。在本文中，我们研究了加权散度形式的椭圆算子方程组的狄利克雷特征值问题 $$\left\{ {\begin{array}{*{20}{l}} {{\mathbb{L} _{_{A,\varphi }}}\text{u} + \alpha [\nabla (\text{div}\text{u}) - \nabla {\varphi }\text{div}\text{u }] = -\varsigma \text{u},\;\;in\;\;\Omega ,} \\ {u{|_{\partial \Omega }} = 0,} \end{array}} \ right.$$ 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 是一个非负常数，u 是一个向量值函数。建立了该问题特征值的一些普遍不等式。而且，作为这些结果的应用，我们根据前 k 个特征值对 ?k+1 的上界和 ?k+1 -?k 的差距给出了一些估计。我们的结果包含 Lame 系统和漂移拉普拉斯算子方程组的一些结果。