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The Krein-von Neumann extension and its connection to an abstract buckling problem. (English) Zbl 1183.35100

Summary: 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, \(S\geq \varepsilon I_{\mathcal H}\) for some \(\varepsilon >0\) in a Hilbert space \(\mathcal H\) to an abstract buckling problem operator.
In the concrete case where \(S = \overline{-\Delta |_{C_0^\infty (\Omega)}}\) in \(L^{2}(\Omega ; d^nx)\) 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 the buckling of a clamped plate, \[ (-\Delta )^{2}u =\lambda (-\Delta )u \text{ in } \Omega ,\quad \lambda \neq 0,\quad u\in H_{0}^{2}(\Omega ) \] 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.)

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
47A10 Spectrum, resolvent
47F05 General theory of partial differential operators
74K20 Plates
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