1,720,986 research outputs found
Sharp Two-Sided Heat Kernel Estimates of Twisted Tubes and Applications
Full text linkWe 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
Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations
No full textOn positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators
No full text
On the localization of binding for Schrödinger operators and its extension to elliptic operators
No full textRepresentation theorems for positive solutions of parabolic equations
No full textWe determine all the minimal positive solutions of the parabolic equation
L
u
=
0
Lu = 0
in
R
n
×
R
−
{{\mathbf {R}}^n} \times {{\mathbf {R}}_ - }
, where
L
L
has time independent coefficients or
L
L
has periodic coefficients in
x
1
,
…
,
x
n
{x_1}, \ldots ,{x_n}
and
t
t
.</p
- …
