# Sharp two-sided heat kernel estimates of twisted tubes and applications

created by kovarik on 29 Aug 2011
modified on 28 Apr 2015

[BibTeX]

Published Paper

Inserted: 29 aug 2011
Last Updated: 28 apr 2015

Journal: Arch. Ration. Mech. Anal.
Volume: 213
Pages: 28
Year: 2014

Abstract:

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian in locally twisted three-dimensional tubes $\Omega$. In particular, we show that for any fixed $x$ the heat kernel decays for large times as $\mathrm{e}^{-E_1t}\, t^{-3/2}$, 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.