Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness

Ranošová, Jarmila

Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 4, pp. 707-723

Résumé

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.

Détail

Classification : 31B10, 35B05, 35K05, 35K10, 35K15
Keywords: heat equation; parabolic function; Weierstrass kernel; set of determination; Harnack inequality; coparabolic thinness; coparabolic minimal thinness; heat ball