Weak boundary components in \(R^ N\) (Q793868)

Given a domain D in \(R^ N\) where \(N\geq 3\), let E be a compact set such that \(\gamma =\partial E\) is a subboundary of D and let \(c_{\gamma}\) be the capacity of \(\gamma\) in the sense of Sario. The author answers two questions: (i) there exists a domain D and \(\gamma\) consisting of one point such that \(c_{\gamma}\neq 0\); (ii) when \(\gamma\) is an isolated subboundary, \(c_{\gamma}=0\) if and only if the Newtonian capacity of E equals zero. The author also gives a characterization of the condition \(c_{\gamma}=0\) in terms of extremal length of order 2, thus extending to \(N\geq 3\) the result of Jurchescu in the plane.