Removable sets for the flux of continuous vector fields

de Valeriola, Sébastien; Moonens, Laurent

doi:10.1090/S0002-9939-09-10092-8

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 $\mathbb {R}^n$.

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