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

 

Removable sets for the flux of continuous vector fields


Authors: Sébastien de Valeriola and Laurent Moonens
Journal: Proc. Amer. Math. Soc. 138 (2010), 655-661
MSC (2000): Primary 49Q15; Secondary 35B60
Published electronically: October 6, 2009
MathSciNet review: 2557182
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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$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10092-8
PII: S 0002-9939(09)10092-8
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.