Lower bounds for the Dirichlet heat kernel

1997 ◽  
Vol 48 (190) ◽  
pp. 203-211
Author(s):  
G Grillo
Keyword(s):  
Heat Kernel ◽  
Lower Bounds ◽  
Dirichlet Heat Kernel

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

Heat Kernels of Lorentz Cones

1999 ◽  
Vol 42 (2) ◽  
pp. 169-173 ◽  
Author(s):  
Hongming Ding
Keyword(s):  
Heat Kernel ◽  
Explicit Formula ◽  
Lower Bounds ◽  
Heat Kernels ◽  
Upper And Lower Bounds ◽  
Symmetric Cones ◽  
Lorentz Cone

AbstractWe obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time t and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.

scholarly journals Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities

2018 ◽  
Vol 2018 (736) ◽  
pp. 1-31
Author(s):  
Alexander Grigor’yan ◽  
Shunxiang Ouyang ◽  
Michael Röckner
Keyword(s):  
Heat Kernel ◽  
Isoperimetric Inequality ◽  
Lower Bounds ◽  
Isoperimetric Inequalities ◽  
Upper And Lower Bounds ◽  
Radial Part ◽  
Kernel Estimates ◽  
Heat Kernel Estimates ◽  
Singular Drift

AbstractIn the present paper we prove upper and lower bounds of the heat kernel for the operator{\Delta-\nabla({|x|^{-\alpha}})\cdot\nabla}in{\mathbb{R}^{n}\setminus\{0\}}, where{\alpha>0}. We obtain these bounds from an isoperimetric inequality for a measure{\mathrm{e}^{-{|x|^{-\alpha}}}dx}on{\mathbb{R}^{n}\setminus\{0\}}. The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.

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