L1-estimates for eigenfunctions of the Dirichlet Laplacian

Michiel van den Berg, Rainer Hempel, Jurgen Voigt

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
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Abstract

For dN and Ω ≠ ∅ an open set in Rd, we consider the eigenfunctions φ of the Dirichlet Laplacian -ΔΩ of Ω. If φ is associated with an eigenvalue below the essential spectrum of -ΔΩ we provide estimates for the L1-norm of Φ in terms of its L2-norm and spectral data. These L1-estimates are then used in the comparison of the heat content of Ω at time t > 0 and the heat trace at times t' > 0, where a two-sided estimate is established. We furthermore show that all eigenfunctions of -ΔΩ which are associated with a discrete eigenvalue of HΩ, belong to L1(Omega).
Original languageEnglish
Pages (from-to)829-857
Number of pages29
JournalJournal of Spectral Theory
Volume5
Issue number4
DOIs
Publication statusPublished - Dec 2015

Keywords

  • Dirichlet Laplacian
  • eigenfunctions
  • L1-estimates
  • heat trace
  • heat content

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