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Keywords:
heat equation; parabolic function; Weierstrass kernel; set of determination; Harnack inequality; coparabolic thinness; coparabolic minimal thinness; heat ball
Summary:
Let $T$ be a positive number or $+\infty$. We characterize all subsets $M$ of $\Bbb R^n \times ]0,T[$ such that $$\inf\limits_{X\in \Bbb R^n \times ]0,T[}u(X) = \inf\limits_{X\in M}u(X) \tag{i}$$ for every positive parabolic function $u$ on $\Bbb R^n \times ]0,T[$ in terms of coparabolic (minimal) thinness of the set $M_\delta =\cup_{(x,t)\in M} B^p((x,t),\delta t)$, where $\delta \in (0,1)$ and $B^p((x,t),r)$ is the heat ball'' with the center'' $(x,t)$ and radius $r$. Examples of different types of sets which can be used instead of heat balls'' are given. It is proved that (i) is equivalent to the condition $\sup_{X\in \Bbb R^n \times \Bbb R^+}u(X) = \sup_{X\in M}u(X)$ for every bounded parabolic function on $\Bbb R^n \times \Bbb R^+$ and hence to all equivalent conditions given in the article [7]. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References.
References:
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