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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 ..."
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Cited by 27 (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
Graphs between the elliptic and parabolic Harnack inequalities.
"... We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L ..."
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Cited by 7 (0 self)
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We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L 2 PoincarĂ© inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass, see [2]. It satisfies the volume regularity and not the PoincarĂ© inequality. We construct another example that does not satisfy the volume regularity.
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.