Analysis of the laplacian and spectral operators on the vicsek set

Sarah Constantin, Robert S. Strichartz, Miles Wheeler

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

Abstract

We study the spectral decomposition of the Laplacian on a family of fractals VSn that includes the Vicsek set for n = 2, extending earlier research on the Sierpinski Gasket. We implement an algorithm [23] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann boundary conditions. We study gaps in the ratios of eigenvalues and eigenvalue clusters. We give an explicit formula for the Green's function on VSn. Finally, we explain how the spectrum of the Laplacian on VSn converges as n ! 1 to the spectrum of the Laplacian on two crossed lines (the limit of the sets VSn).

Original languageEnglish
Pages (from-to)1-44
Number of pages44
JournalCommunications on Pure and Applied Analysis
Volume10
Issue number1
DOIs
Publication statusPublished - 1 Jan 2011

Keywords

  • Eigenvalue clusters
  • Eigenvalue ratio gaps
  • Green's function
  • Heat kernels
  • Laplacians on fractals
  • Spectral operators
  • Vicsek set
  • Wave propagators
  • Weyl ratio

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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