Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Krein-von Neumann Extension and its Connection to an Abstract
  Buckling Problem   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.).

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The categories of this paper are: nlin
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The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph