A Survey on the Krein-von Neumann extension, the corresponding abstract buckling problem, and Weyl-type spectral asymptotics for perturbed Krein Laplacians in nonsmooth domains

Academic Article

Abstract

  • In the first (and abstract) part of this survey 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 ≥ εIH for some ε > 0 in a Hilbert space H to an abstract bucklingpr oblem operator. In the concrete case where S = -Δ|C0∞(Ω) in L2(Ω;dn x) for Ω ⊂ Rn an open, bounded (and sufficiently regular) set, 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 ∈ H20(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.
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    Author List

  • Ashbaugh MS; Gesztesy F; Mitrea M; Shterenberg R; Teschl G
  • Start Page

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  • End Page

  • 106