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On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
Abstract

Cited by 26 (3 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Divergence Form Operators on FractalLike Domains
, 2000
"... We consider elliptic operators L in divergence form on certain domains in R d with fractal volume growth. The domains we look at are preSierpinski carpets, which are derived from higher dimensional Sierpinski carpets. We prove a Harnack inequality for nonnegative Lharmonic functions on these domai ..."
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Cited by 4 (1 self)
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We consider elliptic operators L in divergence form on certain domains in R d with fractal volume growth. The domains we look at are preSierpinski carpets, which are derived from higher dimensional Sierpinski carpets. We prove a Harnack inequality for nonnegative Lharmonic functions on these domains and establish upper and lower bounds for the corresponding heat equation.