## G. Grillo - H. Kovarik - Y. Pinchover

# 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.

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