Estimates of eigenvalues of elliptical differential problems in divergence form

TL;DR

This paper provides universal bounds for eigenvalues of elliptic differential operators in divergence form, including second and fourth-order problems, with applications to eigenvalue gaps and upper bounds.

Contribution

It introduces new universal estimates for eigenvalues of coupled elliptic systems and fourth-order problems, extending known results to broader classes.

Findings

Universal eigenvalue estimates for elliptic systems

Bounds on eigenvalue gaps and individual eigenvalues

Applicable to Lamé, Laplacian, and bi-Laplacian operators

Abstract

In this paper, we compute universal estimates of eigenvalues for a class of coupled systems of elliptic differential equations in divergence form on a bounded domain in Euclidean space, which includes the well-known Lam\'e and the Laplacian operator. Furthermore, we also give universal estimates of eigenvalues for a class of fourth-order elliptic differential problems in divergence form, which encloses the well-known bi-Laplacian operator. In both cases, as applications, we obtain the gap between consecutive eigenvalues as well as an upper bound for each eigenvalue.