Archive for Rational Mechanics and Analysis

, Volume 213, Issue 1, pp 215–243 | Cite as

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

  • Gabriele GrilloEmail author
  • Hynek Kovařík
  • Yehuda Pinchover


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 \({-\Delta^D_\Omega-V}\) .


Heat Kernel Sobolev Inequality Hardy Inequality Straight Tube Brownian Bridge 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Gabriele Grillo
    • 1
    Email author
  • Hynek Kovařík
    • 2
  • Yehuda Pinchover
    • 3
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.DICATAM, Sezione di MatematicaUniversità degli studi di BresciaBresciaItaly
  3. 3.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

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