On the spectral asymptotics for the buckling problem

TL;DR

This paper proves Weyl's law for buckling eigenvalues of the biharmonic operator on various domains, using sharp bounds and variational principles, and confirms a conjecture in special cases.

Contribution

It provides a direct proof of Weyl's law for buckling eigenvalues on broad classes of domains, including new bounds and verification of a conjecture in specific cases.

Findings

Established asymptotically sharp bounds for the Riesz mean $R_2(z)$.

Proved Weyl's law for buckling eigenvalues on Lipschitz domains.

Confirmed the conjecture for balls and bounded intervals.

Abstract

We provide a direct proof of Weyl's law for the buckling eigenvalues of the biharmonic operator on a wide class of domains of $\mathbb R^d$ including bounded Lipschitz domains. The proof relies on asymptotically sharp lower and upper bounds that we develop for the Riesz mean $R_2(z)$. Lower bounds are obtained by making use of the so-called "averaged variational principle". Upper bounds are obtained in the spirit of Berezin-Li-Yau. Moreover, we state a conjecture for the second term in Weyl's law and prove its correctness in two special cases: balls in $\mathbb R^d$ and bounded intervals in $\mathbb R$.