Capacity density of subanalytic sets in higher dimension
This paper concerns the relationship between capacity and subanalytic geometry. Let \(B_{r}(x)\) denote the open ball of centre \(x\) and radius \(r\) in \(\mathbb{R}^{n}\) \((n\geq 3)\), and let \(c(\cdot )\) denote Newtonian capacity. The main result says that, if \(E\subset \mathbb{R}^{n}\) is a subanalytic set, then the capacity density \(\lim_{r\rightarrow 0}c(E\cap B_{r}(x))/c(B_{r}(x))\) exists at every point \(x\in \mathbb{R}^{n}\). The author [Ill. J. Math. 49, 719--736 (2005; Zbl 1092.31004)] had previously established the corresponding result for two dimensions, but the proof of the higher dimensional case required new ideas.
scientific article; zbMATH DE number 5162167