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For an arbitrary open, nonempty, bounded set $\Omega \subset \mathbb{R}^n$,$n \in \mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we considerthe closed, strictly positive, higher-order differential operator $A_{\Omega,2m} (a,b,q)$ in $L^2(\Omega)$ defined on $W_0^{2m,2}(\Omega)$, associated withthe higher-order differential expression $$ \tau_{2m} (a,b,q) :=\bigg(\sum_{j,k=1}^{n} (-i \partial_j - b_j) a_{j,k} (-i \partial_k -b_k)+q\bigg)^m, \quad m \in \mathbb{N}, $$ and its Krein--von Neumann extension$A_{K, \Omega, 2m} (a,b,q)$ in $L^2(\Omega)$. Denoting by $N(\lambda; A_{K,\Omega, 2m} (a,b,q))$, $\lambda > 0$, the eigenvalue counting functioncorresponding to the strictly positive eigenvalues of $A_{K, \Omega, 2m}(a,b,q)$, we derive the bound $$ N(\lambda; A_{K, \Omega, 2m} (a,b,q)) \leq Cv_n (2\pi)^{-n} \bigg(1+\frac{2m}{2m+n}\bigg)^{n/(2m)} \lambda^{n/(2m)} , \quad\lambda > 0, $$ where $C = C(a,b,q,\Omega)>0$ (with $C(I_n,0,0,\Omega) =|\Omega|$) is connected to the eigenfunction expansion of the self-adjointoperator $\widetilde A_{2m} (a,b,q)$ in $L^2(\mathbb{R}^n)$ defined on$W^{2m,2}(\mathbb{R}^n)$, corresponding to $\tau_{2m} (a,b,q)$. Here $v_n :=\pi^{n/2}/\Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in$\mathbb{R}^n$.  Our method of proof relies on variational considerations exploiting thefundamental link between the Krein--von Neumann extension and an underlyingabstract buckling problem, and on the distorted Fourier transform defined interms of the eigenfunction transform of $\widetilde A_{2} (a,b,q)$ in$L^2(\mathbb{R}^n)$.  We also consider the analogous bound for the eigenvalue counting function forthe Friedrichs extension $A_{F,\Omega, 2m} (a,b,q)$ in $L^2(\Omega)$ of$A_{\Omega, 2m} (a,b,q)$.  No assumptions on the boundary $\partial \Omega$ of $\Omega$ are made.

A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions