Article

**Math. Nachr. 283:2, 165-179 (2010)**[DOI: 10.1002/mana.200910067]

## The Krein-von Neumann Extension and its Connection to an Abstract Buckling Problem

### Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, Roman Shterenberg, and Gerald Teschl

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,
is in one-to-one correspondence with the problem of
where
with

*S≥ ε I*for some_{H}*ε >0*in a Hilbert space*H*to an abstract buckling problem operator.
In the concrete case where *S=-Δ| _{C0∞(Ω)}* in

*L*for

^{2}(Ω; d^{n}x)*Ω⊂ℝ*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

^{n}*S*(i.e., the Krein-von Neumann extension of

_{K}*S*),

*S*

_{K}v = λ v, λ ≠ 0,*the buckling of a clamped plate*,*(-Δ)*in

^{2}u=λ (-Δ) u*Ω, λ ≠ 0, u∈ H*

_{0}^{2}(Ω),*u*and*v*are related via the pair of formulas*u = S*

_{F}^{-1}(-Δ) v, v = λ^{-1}(-Δ) u,*S*the Friedrichs extension of_{F}*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*

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