JournalsjstVol. 5, No. 4pp. 829–857

L1L_1-estimates for eigenfunctions of the Dirichlet Laplacian

  • Michiel van den Berg

    University of Bristol, UK
  • Rainer Hempel

    Technische Universität Braunschweig, Germany
  • Jürgen Voigt

    Technische Universität Dresden, Germany
$L_1$-estimates for eigenfunctions of the Dirichlet Laplacian cover
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For dNd \in \mathbb N and Ω\Omega \ne \emptyset an open set in Rd\mathbb R^d, we consider the eigenfunctions Φ\Phi of the Dirichlet Laplacian ΔΩ-\Delta_\Omega of Ω\Omega. If Φ\Phi is associated with an eigenvalue below the essential spectrum of ΔΩ-\Delta_\Omega we provide estimates for the L1L_1-norm of Φ\Phi in terms of its L2L_2-norm and spectral data. These L1L_1-estimates are then used in the comparison of the heat content of Ω\Omega at time t>0t>0 and the heat trace at times t>0t' > 0, where a two-sided estimate is established. We furthermore show that all eigenfunctions of ΔΩ-\Delta_\Omega which are associated with a discrete eigenvalue of HΩH_\Omega, belong to L1(Ω)L_1(\Omega).

Cite this article

Michiel van den Berg, Rainer Hempel, Jürgen Voigt, L1L_1-estimates for eigenfunctions of the Dirichlet Laplacian. J. Spectr. Theory 5 (2015), no. 4, pp. 829–857

DOI 10.4171/JST/115