On the spectral asymptotics for the buckling problem

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 $\mathbb{R}$.

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Cited by in corpus (1)

• On the Weyl Asymptotic Formula for Euclidean Domains of Finite Volume