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

TL;DR

This paper establishes sharp two-sided heat kernel estimates for twisted tubes, demonstrating how twisting accelerates heat decay and applying these results to Sobolev inequalities and spectral analysis of Schrödinger operators.

Contribution

It provides the first precise heat kernel bounds for twisted tubes and shows how twisting influences heat decay and spectral properties.

Findings

Heat kernel decays as e^{-E_1 t} t^{-3/2} in twisted tubes

Twisting speeds up heat decay compared to straight tubes

Applications include Sobolev inequalities and spectral estimates

Abstract

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $-\Delta^D_\Omega$ 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 for Schr\"odinger operators $-\Delta^D_\Omega-V$.