The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in Rd does not always remain unaltered during the flex

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Abstract

Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in special bounded domains of Euclidean d-space. The boundary of such a domain is an embedded simplicial complex which allows a continuous deformation (a flex), under which each simplex of the complex moves as a solid body and the change in the spatial shape of the domain is achieved through a change of the dihedral angles only. The main result of this article is that both the Dirichlet and Neumann spectra of the Laplace operator in such a domain do not necessarily remain unaltered during the flex of its boundary.

Original languageEnglish
Article number32
Number of pages14
JournalJournal of Geometry
Volume111
Issue number2
DOIs
Publication statusPublished - 3 Jun 2020

Keywords

  • Asymptotic behavior of eigenvalues
  • Dihedral angle
  • Dirichlet eigenvalue
  • Flexible polyhedron
  • Laplace operator
  • Neumann eigenvalue
  • Volume
  • Weyl asymptotic formula for the Laplacian
  • Weyl’s law
  • BELLOWS CONJECTURE
  • CROSS-POLYTOPES
  • VOLUME
  • INVARIANT
  • HEAT-EQUATION
  • Weyl's law

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