Estimates of eigenvalues of an elliptic differential system in divergence form

TL;DR

This paper derives universal eigenvalue estimates for elliptic differential systems in divergence form, exploring their invariance under perturbations and applications to Gaussian solitons and divergence-free tensors.

Contribution

It introduces new universal eigenvalue bounds for coupled elliptic systems and examines their invariance properties in Gaussian soliton settings.

Findings

Eigenvalue estimates are invariant under first-order perturbations.

Rigidity inequalities for Laplacian eigenvalues are established in Gaussian shrinking solitons.

Special case analysis for divergence-free tensors related to Cheng-Yau operator.

Abstract

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.