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

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 $(- \Delta)^m\big|_{C_0^{\infty}(\Omega)}$, $m \in \mathbb{N}$, 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 $$N(\lambda,A_{K,\Omega,m}) \leq (2 \pi)^{-n} v_n |\Omega| \{1 + [2m/(2m+n)]\}^{n/(2m)} \lambda^{n/(2m)}, \quad \lambda > 0,$$ 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.Comment: 22 pages. Considerable improvements mad

Understanding linear measure

This article provides strategies for enhancing tasks to offer students better opportunities to develop conceptual understanding of length measurement. Teachers are offered strategies that help move instruction beyond procedures