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

For an arbitrary nonempty, open set $Ω\subset \mathbb{R}^n$, $n \in \mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(- Δ)^m\big|_{C_0^{\infty}(Ω)}$, $m \in \mathbb{N}$, and its Krein--von Neumann extension $A_{K,Ω,m}$ in $L^2(Ω)$. With $N(λ,A_{K,Ω,m})$, $λ> 0$, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,Ω,m}$, we derive the bound $N(λ,A_{K,Ω,m}) \leq (2 π)^{-n} v_n |Ω| \{1 + [2m/(2m+n)]\}^{n/(2m)} λ^{n/(2m)}, \quad λ> 0,$ where $v_n := π^{n/2}/Γ((n+2)/2)$ 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.

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