Jarmila Rano\v{s}ov\'a
Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness

Comment.Math.Univ.Carolinae 37,4 (1996) 707-723.

Abstract: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. \par 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]. \par The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References.

Keywords: heat equation, parabolic function, Weierstrass kernel, set of determination, Harnack inequality, coparabolic thinness, coparabolic minimal thinness, heat ball
AMS Subject Classification: 35K05, 35K15, 31B10

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