Inequalities between Dirichlet and Neumann eigenvalues of the Polyharmonic operators

TL;DR

This paper establishes a strict inequality between Neumann and Dirichlet eigenvalues of polyharmonic operators, revealing fundamental spectral differences under different boundary conditions.

Contribution

It proves a novel inequality relating Neumann and Dirichlet eigenvalues for polyharmonic operators, advancing understanding of spectral properties under boundary conditions.

Findings

Neumann eigenvalues are strictly less than Dirichlet eigenvalues for the same order

The inequality holds for eigenvalues of $(- riangle)^m$ on bounded domains

Provides new insights into spectral inequalities for higher-order operators

Abstract

We prove that $\mu_{k+m}^m <\lambda_k^m$, where $\mu_k^m$ ($\lambda_k^m$) are the eigenvalues of $(-\Delta)^m$ on $\Omega\subset\mathbb R^d$, $d\geq 2$, with Neumann (Dirichlet) boundary conditions.