Abstract
For d ∈ N 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 language | English |
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Pages (from-to) | 829-857 |
Number of pages | 29 |
Journal | Journal of Spectral Theory |
Volume | 5 |
Issue number | 4 |
DOIs | |
Publication status | Published - Dec 2015 |
Keywords
- Dirichlet Laplacian
- eigenfunctions
- L1-estimates
- heat trace
- heat content
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Professor Michiel van den Berg
- School of Mathematics - Emeritus Professor
- Probability, Analysis and Dynamics
- Pure Mathematics
- Analysis
Person: Academic , Member, Honorary and Visiting Academic