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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