Estimates of eigenvalues of an elliptic differential system in divergence form

Abstract

In this paper, we compute universal estimates of eigenvalues of a coupled system of elliptic differential equations in divergence form in 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.Comment: 23 page