Mattila, P.; Paramonov, P. V. On geometric properties of harmonic \(\text{Lip}_ 1\)-capacity. (English) Zbl 0852.31004 Pac. J. Math. 171, No. 2, 469-491 (1995). 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 \({\mathcal H}^{n - 1} (E) > 0\). Reviewer: P.Mattila (Jyväskylä) Cited in 26 Documents MSC: 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31B15 Potentials and capacities, extremal length and related notions in higher dimensions 28A75 Length, area, volume, other geometric measure theory Keywords:harmonic \(\text{Lip}_ 1\)-capacity; harmonic functions; capacities; Hausdorff measures; geometric properties PDFBibTeX XMLCite \textit{P. Mattila} and \textit{P. V. Paramonov}, Pac. J. Math. 171, No. 2, 469--491 (1995; Zbl 0852.31004) Full Text: DOI