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

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, $S\ge ϵI_{\mathcal{H}}$ for some $ϵ>0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. In the concrete case where $S=\stackrel{ˉ}{-\mathrm{\Delta }{\mathrm{\mid }}_{C_0^{\mathrm{\infty }}(\mathrm{\Omega })}}$ in $L^2(\mathrm{\Omega };d^{n}x)$ for $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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 $S$), $$S_{K}v=\lambda v,\lambda \mathrm{\ne }0,$$ is in one-to-one correspondence with the problem of {\em the buckling of a clamped plate}, $$(-\mathrm{\Delta }{)}^{2}u=\lambda (-\mathrm{\Delta })u\text{in}\mathrm{\Omega },\lambda \mathrm{\ne }0,u\in H_0^2(\mathrm{\Omega }),$$ where $u$ and $v$ are related via the pair of formulas $$u=S_F^{-1}(-\mathrm{\Delta })v,v={\lambda }^{-1}(-\mathrm{\Delta })u,$$ 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.).