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

Let A be a symmetric and positive definite (1, 1) tensor on a bounded domain Ω in an ndimensional metric measure space (Rⁿ, < , > , e^-ψdv). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form where L_A,ψ = div(A?(·))-, 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 ?_k+1 and the gap of ?_k+1 -?_k in terms of the first k eigenvalues. Our results contain some results for the Lamé system and a system of equations of the drifting Laplacian.