Sharp inequalities and asymptotics for polyharmonic eigenvalues

TL;DR

This paper investigates inequalities, growth estimates, and asymptotic behaviors of eigenvalues for polyharmonic operators in various domains, providing new bounds and asymptotic formulas especially for high-order cases.

Contribution

It establishes new inequalities and growth estimates for polyharmonic eigenvalues, including precise asymptotics for the first eigenvalue as the order increases.

Findings

Derived inequalities for eigenvalues on general domains.

Obtained bounds for the first eigenvalue in the ball.

Established asymptotic expansion for the first eigenvalue as order grows.

Abstract

We study eigenvalues of general scalar Dirichlet polyharmonic problems in domains in $\mathbb R^{d}$. We first prove a number of inequalities satisfied by the eigenvalues on general domains, depending on the relations between the orders of the operators involved. We then obtain several estimates for these eigenvalues, yielding their growth as a function of these orders. For the problem in the ball we derive the general form of eigenfunctions together with the equations satisfied by the corresponding eigenvalues, and obtain several bounds for the first eigenvalue. In the case of the polyharmonic operator of order $2m$ we derive precise bounds yielding the first two terms in the asymptotic expansion for the first normalised eigenvalue as $m$ grows to infinity. These results allow us to obtain the order of growth for the $k^{\rm th}$ polyharmonic eigenvalue on general domains.