We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian −Δ^D_Ω in locally twisted three-dimensional tubes Ω. In particular, we show that for any fixed x the heat kernel decays for large times as e^{−E_1}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 −Δ^D_Ω−V .

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

GRILLO, GABRIELE;
2014-01-01

Abstract

We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian −Δ^D_Ω in locally twisted three-dimensional tubes Ω. In particular, we show that for any fixed x the heat kernel decays for large times as e^{−E_1}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 −Δ^D_Ω−V .
Heat kernel; asymptotics.; subcritical operators; spectral bounds; Sobolev and Nash inequalities
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/804118
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