Some New Inequalities of Dirichlet Eigenvalues for Laplace Operator with any Order

TL;DR

This paper derives new, more general inequalities for Dirichlet eigenvalues of the Laplace operator of any order in n-dimensional Euclidean space, extending previous results with novel mathematical techniques.

Contribution

It introduces a broader class of inequalities for Dirichlet eigenvalues, surpassing existing Yang's inequalities, using a generalized Chebyshev's inequality approach.

Findings

Established new inequalities for Dirichlet eigenvalues

Extended Yang's inequalities to more general cases

Provided new mathematical consequences of these inequalities

Abstract

In this paper, we establish several inequalities of Dirichlet eigenvalues for Laplace operator $\Delta $ with any order on \emph{n}-dimensional Euclidean space. These inequalities are more general than known Yang's inequalities and contain new consequences. To obtain them, we borrow the approach of Illias and Makhoul, and use a generalized Chebyshev's inequality.