Removable sets for the flux of continuous vector fields
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- by Sébastien de Valeriola and Laurent Moonens PDF
- Proc. Amer. Math. Soc. 138 (2010), 655-661 Request permission
Abstract:
We show that any closed set $E$ having a $\sigma$-finite $(n-1)$-dimensional Hausdorff measure does not support the nonzero distributional divergence of a continuous vector field; in particular it has the property that any $C^1$ function in $\mathbb {R}^n$ that is harmonic outside it is harmonic in $\mathbb {R}^n$. We also exhibit a compact set $E$ having Hausdorff dimension $n-1$, supporting the nonzero distributional divergence of a continuous vector field yet having the property that any $C^1$ function that is harmonic outside $E$ is harmonic in $\mathbb {R}^n$.References
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Additional Information
- Sébastien de Valeriola
- Affiliation: Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron, 2, 1348 Louvain-la-neuve, Belgium
- Email: sebastien.devaleriola@uclouvain.be
- Laurent Moonens
- Affiliation: Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron, 2, 1348 Louvain-la-neuve, Belgium
- Email: laurent.moonens@uclouvain.be
- Received by editor(s): January 7, 2009
- Received by editor(s) in revised form: June 10, 2009
- Published electronically: October 6, 2009
- Additional Notes: The second author is an aspirant of the Fonds de la Recherche scientifique — FNRS (Belgium).
- Communicated by: Tatiana Toro
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 655-661
- MSC (2000): Primary 49Q15; Secondary 35B60
- DOI: https://doi.org/10.1090/S0002-9939-09-10092-8
- MathSciNet review: 2557182