Remarks on the results by Koskela concerning the radial uniqueness for Sobolev functions
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- by Yoshihiro Mizuta PDF
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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 \[ \int _{U(\varepsilon )} |\nabla u(x)|^p dx \leqq \varepsilon ^p \varphi (\varepsilon ) \hspace {2em} \text {for any $\varepsilon > 0$,} \] 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.References
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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
- Received by editor(s): September 18, 1996
- Communicated by: Albert Baernstein II
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 1043-1047
- MSC (1991): Primary 31B25, 31B15, 46E35
- DOI: https://doi.org/10.1090/S0002-9939-98-04296-8
- MathSciNet review: 1443397