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Graphs between the elliptic and parabolic Harnack inequalities. (English) Zbl 1081.39012
Summary: 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 that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a $$L^2$$ Poincaré inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass. It satisfies the volume regularity and not the Poincaré inequality. We construct another example that does not satisfy the volume regularity.

##### MSC:
 39A12 Discrete version of topics in analysis 31C20 Discrete potential theory 35J15 Second-order elliptic equations
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