The Krein–von Neumann extension and its connection to an abstract buckling problem

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)