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

✍ Scribed by Mark S. Ashbaugh; Fritz Gesztesy; Marius Mitrea; Roman Shterenberg; Gerald Teschl

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 193 KB
• Volume: 283
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200910067

✦ Synopsis

Abstract

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 ≥ εI~H~ for some ε > 0 in a Hilbert space H to an abstract buckling problem operator.

In the concrete case where in L^2^(Ω; d^n^x) for Ω ⊂ 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 = λ~v~, λ ≠ 0,
is in one‐to‐one correspondence with the problem of the buckling of a clamped plate,
(‐Δ)^2^u = λ (‐Δ)u in Ω, λ ≠ 0, u ∈ H~0~^2^(Ω),
where u and v are related via the pair of formulas
u = S~F~^‐1^ (‐Δ)v, v = λ^‐1^(‐Δ)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.) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)