Lower bounds for eigenvalues of Laplacian operator and the clamped plate problem

TL;DR

This paper establishes sharp lower bounds for eigenvalues of the Laplace operator and the clamped plate problem, extending previous estimates to more general cases and arbitrary dimensions.

Contribution

It provides new sharp lower bounds for eigenvalues of both the Laplace operator and the clamped plate problem, including general and higher-dimensional cases.

Findings

Sharp lower bounds for Laplace eigenvalues proved.

Extended eigenvalue estimates to general cases.

Derived bounds for the clamped plate problem in arbitrary dimensions.

Abstract

In this paper, we give some lower bounds for several eigenvalues. Firstly, we investigate the eigenvalues $\lambda_i$ of the Laplace operator and prove a sharp lower bound. Moreover, we extent this estimate of the eigenvalues to general cases. Secondly, we study the eigenvalues $\Gamma_i$ for the clamped plate problem and deliver a sharp bound for the clamped plate problem for arbitrary dimension.