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Sharp Two-Sided Heat Kernel Estimates of Twisted Tubes and Applications

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Abstract

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian \({-\Delta^D_\Omega}\) in locally twisted three-dimensional tubes Ω. In particular, we show that for any fixed x the heat kernel decays for large times as \({{\rm e}^{-E_1t} t^{-3/2}}\) , where E 1 is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined on a straight tube. We also discuss some applications of this result, such as Sobolev inequalities and spectral estimates for Schrödinger operators \({-\Delta^D_\Omega-V}\) .

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Gabriele Grillo

  2. DICATAM, Sezione di Matematica, Università degli studi di Brescia, Via Branze, 38, 25123, Brescia, Italy

    Hynek Kovařík

  3. Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Yehuda Pinchover

Authors
  1. Gabriele Grillo
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  2. Hynek Kovařík
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  3. Yehuda Pinchover
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Correspondence to Gabriele Grillo.

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Communicated by The Editors

G.G. has been partially supported by the MIUR-PRIN 2009 grant “Metodi di viscosità, geometrici e di controllo per modelli diffusivi nonlineari”. H.K. has been partially supported by the MIUR-PRIN 2010-11 grant for the project “Calcolo delle Variazioni”. Y.P. acknowledges the support of the Israel Science Foundation (Grants No. 963/11) founded by the Israel Academy of Sciences and Humanities. G.G. and H.K. acknowledge the support of Gruppo Nazionale per l’AnalisiMatematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Grillo, G., Kovařík, H. & Pinchover, Y. Sharp Two-Sided Heat Kernel Estimates of Twisted Tubes and Applications. Arch Rational Mech Anal 213, 215–243 (2014). https://doi.org/10.1007/s00205-014-0723-3

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  • DOI: https://doi.org/10.1007/s00205-014-0723-3

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