L1-estimates for eigenfunctions of the Dirichlet Laplacian

Michiel van den Berg; Rainer Hempel; Jurgen Voigt

doi:10.4171/JST/115

L₁-estimates for eigenfunctions of the Dirichlet Laplacian

Michiel van den Berg, Rainer Hempel, Jurgen Voigt

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Abstract

For d ∈ N and Ω ≠ ∅ an open set in R^d, we consider the eigenfunctions φ of the Dirichlet Laplacian -Δ_Ω of Ω. If φ is associated with an eigenvalue below the essential spectrum of -Δ_Ω 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 -Δ_Ω which are associated with a discrete eigenvalue of H_Ω, belong to L₁(Omega).

Original language	English
Pages (from-to)	829-857
Number of pages	29
Journal	Journal of Spectral Theory
Volume	5
Issue number	4
DOIs	https://doi.org/10.4171/JST/115
Publication status	Published - Dec 2015

Keywords

Dirichlet Laplacian
eigenfunctions
L1-estimates
heat trace
heat content

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10.4171/JST/115

L1Estimates
This is the author accepted manuscript (AAM). The final published version (version of record) is available online via EMS at http://www.ems-ph.org/journals/show_abstract.php?issn=1664-039X&vol=5&iss=4&rank=6
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Professor Michiel van den Berg
- M.vandenBerg@bristol.ac.uk
- School of Mathematics - Emeritus Professor
- Probability, Analysis and Dynamics
- Pure Mathematics
- Analysis
Person: Academic , Member, Honorary and Visiting Academic

Cite this

@article{fa9c08918ea5462dac9723a4ac529646,

title = "L1-estimates for eigenfunctions of the Dirichlet Laplacian",

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).",

keywords = "Dirichlet Laplacian, eigenfunctions, L1-estimates, heat trace, heat content",

author = "{van den Berg}, Michiel and Rainer Hempel and Jurgen Voigt",

year = "2015",

month = dec,

doi = "10.4171/JST/115",

language = "English",

volume = "5",

pages = "829--857",

journal = "Journal of Spectral Theory",

issn = "1664-039X",

publisher = "European Mathematical Society",

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TY - JOUR

T1 - L1-estimates for eigenfunctions of the Dirichlet Laplacian

AU - van den Berg, Michiel

AU - Hempel, Rainer

AU - Voigt, Jurgen

PY - 2015/12

Y1 - 2015/12

N2 - 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).

AB - 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).

KW - Dirichlet Laplacian

KW - eigenfunctions

KW - L1-estimates

KW - heat trace

KW - heat content

U2 - 10.4171/JST/115

DO - 10.4171/JST/115

M3 - Article (Academic Journal)

VL - 5

SP - 829

EP - 857

JO - Journal of Spectral Theory

JF - Journal of Spectral Theory

SN - 1664-039X

IS - 4

ER -

L1-estimates for eigenfunctions of the Dirichlet Laplacian

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Professor Michiel van den Berg

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L₁-estimates for eigenfunctions of the Dirichlet Laplacian