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.).
Categories: math.SP math.AP