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≥ ε IH
for some ε >0
in a Hilbert space H
to an abstract buckling problem operator.
In the concrete case where S=-Δ|C0∞(Ω) in
L2(Ω; dn x) 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),
SK v = λ v, 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,
are related via the pair of formulas
u = SF-1 (-Δ) v, v = λ-1(-Δ) u,
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.).
MSC2000: Primary 35J25, 35J40, 47A05; Secondary 47A10, 47F05.
Keywords: Krein-von Neumann extension, buckling problem