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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Remarks on the results by Koskela concerning the radial uniqueness for Sobolev functions


Author: Yoshihiro Mizuta
Journal: Proc. Amer. Math. Soc. 126 (1998), 1043-1047
MSC (1991): Primary 31B25, 31B15, 46E35
MathSciNet review: 1443397
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Abstract: In this note we aim to complete the results by Koskela concerning the radial uniqueness for Sobolev functions.

Let $\varphi$ be a positive nonincreasing function on the interval $(0,\infty)$, and let $\mathbf{B}$ denote the unit ball of $R^n$. Consider a $p$-precise function $u$ on $\mathbf{B}$ such that

\begin{displaymath}\int _{U(\varepsilon)} |\nabla u(x)|^p dx \leqq \varepsilon^p \varphi(\varepsilon) \hspace{2em} \text{for any $\varepsilon > 0$,} \end{displaymath}

where $U(\varepsilon) = \{x \in \mathbf{B} : |u(x)|< \varepsilon\}$. We give conditions on $\varphi$ which assure that $u = 0$ whenever $u$ has vanishing fine boundary limits on a set of positive $p$-capacity.

We are also concerned with the sharpness.


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Additional Information

Yoshihiro Mizuta
Affiliation: The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739, Japan
Email: mizuta@mis.hiroshima-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04296-8
PII: S 0002-9939(98)04296-8
Keywords: $p$-precise functions, Sobolev functions, capacity, fine boundary limits
Received by editor(s): September 18, 1996
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1998 American Mathematical Society