Estimates of eigenvalues of an elliptic differential system in divergence form

In this paper, we compute universal estimates of eigenvalues a coupled system elliptic differential equations in divergence form on bounded domain Euclidean space. As an application, show interesting case rigidity inequalities the Laplacian, more precisely, consider countable family domains Gaussian shrinking soliton that makes behavior known Laplacian invariant by first-order perturbation Laplacian. We also address expanding two different settings. finish with special divergence-free tensors which is closely related to Cheng-Yau operator.