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