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

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, $S\geq εI_{\mathcal{H}}$ for some $ε>0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. In the concrete case where $S=\bar{-Δ|_{C_0^\infty(Ω)}}$ in $L^2(Ω; d^n x)$ for $Ω\subset\mathbb{R}^n$ 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 $S$), \[ S_K v = λv, \quad λ\neq 0, \] is in one-to-one correspondence with the problem of {\em the buckling of a clamped plate}, \[ (-Δ)^2u=λ(-Δ) u \text{in} Ω, \quad λ\neq 0, \quad u\in H_0^2(Ω), \] where $u$ and $v$ are related via the pair of formulas \[ u = S_F^{-1} (-Δ) v, \quad v = λ^{-1}(-Δ) u, \] 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.).