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

For an arbitrary nonempty, open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n \in $\mathbb{N}$$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(-\mathrm{\Delta }{)}^m{\mid }_{C_0^{\mathrm{\infty }}(\mathrm{\Omega })}$, $m \in $\mathbb{N}$$,anditsKrein--vonNeumannextension$A_{K,\Omega,m}$in$L^2(\Omega)$\mathrm{.}With$N(\lambda,A_{K,\Omega,m})$,$\lambda > 0$, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,\mathrm{\Omega },m}$, we derive the bound $$N(\lambda ,A_{K,\mathrm{\Omega },m})\le (2\pi {)}^{-n}{v}_n\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\{1+[2m\mathrm{/}(2m+n)]{\}}^{n\mathrm{/}(2m)}{\lambda }^{n\mathrm{/}(2m)},\lambda >0,$$ where $v_n:={\pi }^{n\mathrm{/}2}\mathrm{/}\mathrm{\Gamma }((n+2)\mathrm{/}2)$ 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.