On the lower bounds of the $p$-modulus of families

Abstract

We study the problem of the lower bounds of the modulus of families of paths of order $p,$ $p>n-1,$ and their connection with the geometry of domains containing the specified families. Among other things, we have proved an analogue of N\"akki's theorem on the positivity of the $p$-module of families of paths joining a pair of continua in the given domain. The geometry of domains with a strongly accessible boundary in the sense of the $p$-modulus of families of paths was also studied. We show that domains with a $p$-strongly accessible boundary with respect to a $p$-modulus, $p>n-1,$ are are finitely connected at their boundary. The mentioned result generalizes N\"akki's result, which was proved for uniform domains in the case of a conformal modulus.