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

Academic Article


  • 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{script} for some ε > 0 in a Hilbert space H{script} to an abstract buckling problem operator. In the concrete case where for Ω ⊂ R{double-struck}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 SK (i.e., the Krein-von Neumann extension of S), is in one-to-one correspondence with the problem of the buckling of a clamped plate, where u and v are related via the pair of formulas, 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.
  • Digital Object Identifier (doi)

    Pubmed Id

  • 9761830
  • Author List

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

  • 165
  • End Page

  • 179
  • Volume

  • 283
  • Issue

  • 2