Removable Sets for the Flux of Continuous Vector Fields

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 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