我们证明了一个稠密定义的，封闭的，严格正算子，$S\geq εI_{\mathcal{H}}$的Krein-von Neumann扩张（在其核的正交补上）的逆的酉等价性，对于Hilbert空间中的一些$ε>0$，$\mathcal{H}$到一个抽象的屈曲问题算子。在具体情况下，其中对于$Ω\subset\mathbb{R}^n$，$L^2(Ω; d^n x)$中的$S=\bar{-Δ|_{C_0^\infty(Ω)}}$是开放的、有界的（并且充分正则的）域，这恢复为由于G. Grubb，Krein Laplacian $S_K$的特征值问题（即，$S$的Krein-von Neumann扩展），\[ S_K v = λv, \quad λ\neq 0, \]与{ \em夹紧板的屈曲}问题，\[ (-Δ)^2u=λ(-Δ) u \text{in} Ω, \quad λ\neq 0, \quad u\in H_0^2(Ω), \]一一对应，其中$u$和$v$通过一对公式\[ u = S_F^{-1} (-Δ) v, \quad v = λ^{-1}(-Δ) u, \]与$S_F$相关，$S$的Friedrich扩展。这建立了克莱因扩张作为弹性理论中的自然对象（类似于弗里德里希扩张，它在量子力学，弹性等中找到了自然的应用）。
英文摘要：
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.).
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
