Removable sets for the flux of continuous vector fields

Abstract

We show that any closed set E having a σ-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 R n that is harmonic outside it is harmonic in 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 R n .

Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates

“…The reverse implication "⇐" has been established by de Valeriola and Moonens [11,Theorem 12]. In Section 2, we provide the direct implication "⇒" by showing that if S is not σ-finite for the Hausdorff measure H N −1 , then there exists a (Borel) positive measure µ supported on S such that the equation div V = µ in R N has a continuous solution.…”

“…Proof of Theorem 1.1. The reverse implication "⇐" is established in [11,Theorem 12]. In order to prove the direct implication "⇒", let S ⊂ R N be a closed set which is not σ-finite for the Hausdorff measure H N −1 .…”