Extremal lengths and Green capacities of condensers (Q919141)

Generally, in the theory of condenser capacity in \({\mathbb{R}}^ p\) the relation \(M_ 2(\Gamma^ 0_ E)\leq M_ 2(\Gamma_ E)\) holds, where \(M_ 2(\Gamma)\) is the 2-modulus of the \(\Gamma\)-family; \[ \Gamma_ E:=\Gamma (E^+,E^-;D)\cup \Gamma_{\omega}(E^+,E^-);\quad \Gamma^ 0_ E:=\Gamma (E^+,E^-;D)\cup \Gamma^ 0_{\omega}(E^+,E^-). \] The above relation is a consequence of the relation \(\Gamma^ 0_ E\subset \Gamma_ E\). In the present paper, the author gives conditions in which the relations: \(M_ 2(\Gamma_ E)=a_ p cap E\) and \(M_ 2(\Gamma^ 0_ E)=M_ 2(\Gamma_ E)=a_ p cap E\) hold, \(a_ p\) being the 2-surface of the unit sphere of \({\mathbb{R}}^ p\) multiplied with p.