The Krein-von Neumann Extension and its Connection to an Abstract Buckling Problem

TL;DR

This paper establishes a connection between the Krein--von Neumann extension of a positive operator and an abstract buckling problem, with concrete implications for the spectral theory of the Krein Laplacian and elasticity.

Contribution

It proves the unitary equivalence of the inverse of the Krein extension to an abstract buckling operator, linking spectral properties to elasticity problems.

Findings

Krein extension inverse is unitarily equivalent to a buckling problem operator.

Eigenvalues of the Krein Laplacian correspond to buckling eigenvalues.

Results generalize known cases in elasticity and spectral theory.

Abstract

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, $S\geq \epsilon I_{\mathcal{H}}$ for some $\epsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. In the concrete case where $S=\bar{-\Delta|_{C_0^\infty(\Omega)}}$ in $L^2(\Omega; d^n x)$ for $\Omega\subset\mathbb{R}^n$ an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian $S_K$ (i.e., the Krein--von Neumann extension of $S$), \[ S_K v = \lambda v, \quad \lambda \neq 0, \] is in one-to-one correspondence with the problem of {\em the buckling of a clamped plate}, \[ (-\Delta)^2u=\lambda (-\Delta) u \text{in} \Omega, \quad \lambda \neq 0, \quad u\in H_0^2(\Omega), \] where $u$ and $v$ are related via the pair of formulas \[ u = S_F^{-1} (-\Delta) v, \quad v = \lambda^{-1}(-\Delta) u, \] with $S_F$ the Friedrichs extension of $S$. 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.).