A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets

Abstract

For an arbitrary nonempty, open set $\Omega \subset {\mathbb{R}}^n$, $n\in \mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(-\Delta {)}^m{|}_{C_0^{\infty }(\Omega )}$, $m\in \mathbb{N}$, and its Krein--von Neumann extension $A_{K,\Omega ,m}$ in $L^2(\Omega )$. With $N(\lambda ,A_{K,\Omega ,m})$, $\lambda >0$, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,\Omega ,m}$, we derive the bound $$N(\lambda ,A_{K,\Omega ,m})\le (2\pi {)}^{-n}{v}_n|\Omega |\{1+[2m/(2m+n)]{\}}^{n/(2m)}{\lambda }^{n/(2m)},\lambda >0,$$ where $v_n:={\pi }^{n/2}/\Gamma ((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in ${\mathbb{R}}^n$. The proof relies on variational considerations and exploits the fundamental link between the Krein--von Neumann extension and an underlying (abstract) buckling problem.

22 pages. Considerable improvements made