On geometric properties of harmonic \(\text{Lip}_ 1\)-capacity (Q1918410)

We study geometric properties of the harmonic \(\text{Lip}_1\)-capacity \(\kappa_n' (E)\), \(E \subset \mathbb{R}^n\). It is related to functions which are harmonic outside \(E\) and locally Lipschitzian everywhere. We show that \(\kappa_{n + 1}' (E \times I)\) is comparable to \(\kappa_n' (E)\) for \(E \subset \mathbb{R}^n\) and for intervals \(I \subset \mathbb{R}\). We also show that if \(E\) lies on a Lipschitz graph, then \(\kappa_n' (E)\) is comparable to the \((n - 1)\)-dimensional Hausdorff measure \({\mathcal H}^{n - 1} (E)\). Finally we give some general criteria to guarantee that \(\kappa_n' (E) = 0\) although \({\mathcal H}^{n - 1} (E) > 0\).