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

Commentationes Mathematicae Universitatis Carolinae (1996)

- Volume: 37, Issue: 4, page 707-723
- ISSN: 0010-2628

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topRanošová, Jarmila. "Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness." Commentationes Mathematicae Universitatis Carolinae 37.4 (1996): 707-723. <http://eudml.org/doc/247931>.

@article{Ranošová1996,

abstract = {Let $T$ be a positive number or $+\infty $. We characterize all subsets $M$ of $\mathbb \{R\}^n \times ]0,T[ $ such that \[ \inf \limits \_\{X\in \mathbb \{R\}^n \times ]0,T[\}u(X) = \inf \limits \_\{X\in M\}u(X) i \]
for every positive parabolic function $u$ on $\mathbb \{R\}^n \times ]0,T[$ 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 \mathbb \{R\}^n \times \mathbb \{R\}^+\}u(X) = \sup _\{X\in M\}u(X) $ for every bounded parabolic function on $\mathbb \{R\}^n \times \mathbb \{R\}^+$ 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.},

author = {Ranošová, Jarmila},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {heat equation; parabolic function; Weierstrass kernel; set of determination; Harnack inequality; coparabolic thinness; coparabolic minimal thinness; heat ball; parabolic function; Harnack inequality},

language = {eng},

number = {4},

pages = {707-723},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness},

url = {http://eudml.org/doc/247931},

volume = {37},

year = {1996},

}

TY - JOUR

AU - Ranošová, Jarmila

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

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1996

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 37

IS - 4

SP - 707

EP - 723

AB - Let $T$ be a positive number or $+\infty $. We characterize all subsets $M$ of $\mathbb {R}^n \times ]0,T[ $ such that \[ \inf \limits _{X\in \mathbb {R}^n \times ]0,T[}u(X) = \inf \limits _{X\in M}u(X) i \]
for every positive parabolic function $u$ on $\mathbb {R}^n \times ]0,T[$ 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 \mathbb {R}^n \times \mathbb {R}^+}u(X) = \sup _{X\in M}u(X) $ for every bounded parabolic function on $\mathbb {R}^n \times \mathbb {R}^+$ 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.

LA - eng

KW - heat equation; parabolic function; Weierstrass kernel; set of determination; Harnack inequality; coparabolic thinness; coparabolic minimal thinness; heat ball; parabolic function; Harnack inequality

UR - http://eudml.org/doc/247931

ER -

## References

top- Aikawa H., Sets of determination for harmonic function in an NTA domains, J. Math. Soc. Japan, to appear. MR1376083
- 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. (1987) Zbl0658.31001MR0908454
- Brzezina M., On the base and the essential base in parabolic potential theory, Czechoslovak Math. J. 40 (115) (1990), 87-103. (1990) Zbl0712.31001MR1032362
- Doob J.L., Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag New York (1984). (1984) Zbl0549.31001MR0731258
- Gardiner S.J., Sets of determination for harmonic function, Trans. Amer. Math. Soc. 338.1 (1993), 233-243. (1993) MR1100694
- Moser J., A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. XVII (1964), 101-134. (1964) Zbl0149.06902MR0159139
- Ranošová J., Sets of determination for parabolic functions on a half-space, Comment. Math. Univ. Carolinae 35 (1994), 497-513. (1994) MR1307276
- Watson A.N., Thermal capacity, Proc. London Math. Soc. 37.3 (1987), 342-362. (1987) MR0507610

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