Abstract
AbstractWe 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 ≥ εIH for some ε > 0 in a Hilbert space H to an abstract buckling problem operator.In the concrete case where in L2(Ω; dnx) for Ω ⊂ ℝn 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 SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ≠ 0, is in one‐to‐one correspondence with the problem of the buckling of a clamped plate, (‐Δ)2u = λ (‐Δ)u in Ω, λ ≠ 0, u ∈ H02(Ω), where u and v are related via the pair of formulas u = SF‐1 (‐Δ)v, v = λ‐1(‐Δ)u, with SF 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.) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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