| Title:
|
Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness (English) |
| Author:
|
Ranošová, Jarmila |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
37 |
| Issue:
|
4 |
| Year:
|
1996 |
| Pages:
|
707-723 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
heat equation |
| Keyword:
|
parabolic function |
| Keyword:
|
Weierstrass kernel |
| Keyword:
|
set of determination |
| Keyword:
|
Harnack inequality |
| Keyword:
|
coparabolic thinness |
| Keyword:
|
coparabolic minimal thinness |
| Keyword:
|
heat ball |
| MSC:
|
31B10 |
| MSC:
|
35B05 |
| MSC:
|
35K05 |
| MSC:
|
35K10 |
| MSC:
|
35K15 |
| idZBL:
|
Zbl 0887.35064 |
| idMR:
|
MR1440703 |
| . |
| Date available:
|
2009-01-08T18:27:30Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118880 |
| . |
| Reference:
|
[1] Aikawa H.: Sets of determination for harmonic function in an NTA domains.J. Math. Soc. Japan, to appear. MR 1376083 |
| Reference:
|
[2] Bonsall F.F.: Domination of the supremum of a bounded harmonic function by its supremum over a countable subset.Proc. Edinburgh Math. Soc. 30 (1987), 441-477. Zbl 0658.31001, MR 0908454 |
| Reference:
|
[3] Brzezina M.: On the base and the essential base in parabolic potential theory.Czechoslovak Math. J. 40 (115) (1990), 87-103. Zbl 0712.31001, MR 1032362 |
| Reference:
|
[4] Doob J.L.: Classical Potential Theory and Its Probabilistic Counterpart.Springer-Verlag New York (1984). Zbl 0549.31001, MR 0731258 |
| Reference:
|
[5] Gardiner S.J.: Sets of determination for harmonic function.Trans. Amer. Math. Soc. 338.1 (1993), 233-243. MR 1100694 |
| Reference:
|
[6] Moser J.: A Harnack inequality for parabolic differential equations.Comm. Pure Appl. Math. XVII (1964), 101-134. Zbl 0149.06902, MR 0159139 |
| Reference:
|
[7] Ranošová J.: Sets of determination for parabolic functions on a half-space.Comment. Math. Univ. Carolinae 35 (1994), 497-513. MR 1307276 |
| Reference:
|
[8] Watson A.N.: Thermal capacity.Proc. London Math. Soc. 37.3 (1987), 342-362. MR 0507610 |
| . |
