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

Jarmila Ranošová

Commentationes Mathematicae Universitatis Carolinae (1996)

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

Abstract

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Let T be a positive number or + ∞ . We characterize all subsets M of ℝ n × ] 0 , T [ such that inf X ∈ ℝ n × ] 0 , T [ u ( X ) = inf X ∈ M u ( X ) i for every positive parabolic function u on ℝ n × ] 0 , T [ in terms of coparabolic (minimal) thinness of the set M δ = ∪ ( x , t ) ∈ M B p ( ( x , t ) , δ t ) , where δ ∈ ( 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 ∈ ℝ n × ℝ + u ( X ) = sup X ∈ M u ( X ) for every bounded parabolic function on ℝ n × 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.

How to cite

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Ranošová, 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

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

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