A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

TL;DR

This survey explores the Krein-von Neumann extension's spectral properties, its connection to buckling problems, and Weyl-type asymptotics for perturbed Laplacians in nonsmooth domains, highlighting its significance in elasticity theory.

Contribution

It establishes the unitary equivalence of the Krein extension's inverse to a buckling problem operator and analyzes spectral properties in nonsmooth domains.

Findings

Krein extension is unitarily equivalent to a buckling problem operator.

Spectral asymptotics for perturbed Krein Laplacians are characterized in nonsmooth domains.

The Krein extension's role in elasticity theory is elucidated.

Abstract

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.