 09109 Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, Roman Shterenberg, and Gerald Teschl
 The Kreinvon Neumann Extension and its Connection to an Abstract Buckling Problem
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Jul 12, 09

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Abstract. We prove the unitary equivalence of the inverse of the Kreinvon 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
$L^2(\Omega; d^n x)$ 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 Kreinvon Neumann extension of $S$),
\[
S_K v = \lambda v, \quad \lambda \neq 0,
\]
is in onetoone 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.).
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