Bounds for eigenvalue ratios of the Laplacian

TL;DR

This paper establishes bounds on the ratios of Laplacian eigenvalues for bounded domains, providing general inequalities and insights into the relationships among lower order eigenvalues.

Contribution

It introduces new inequalities for Laplacian eigenvalues and derives bounds for ratios of lower order eigenvalues in bounded Euclidean domains.

Findings

Derived a general inequality for Laplacian eigenvalues.

Established bounds for ratios of lower order eigenvalues.

Applied inequalities to specific eigenvalue problems.

Abstract

For a bounded domain $\Omega$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Laplacian and derive the ratios of lower order eigenvalues of the Laplacian.