The trace of $BV$-functions on an irregular subset

Abstract:

Certain basic results on the boundary trace discussed in Maz′ya’s monograph on Sobolev spaces are generalized to a wider class of regions. The paper is an extended and supplemented version of a preliminary publication, where some results were presented without proofs or in a weaker form. In Maz′ya’s monograph, the boundary trace was defined for regions $\Omega$ with finite perimeter, and the main results were obtained under the assumption that normals in the sense of Federer exist almost everywhere on the boundary. Instead, now it is assumed that the region boundary is a countably $(n-1)$-rectifiable set, which is a more general condition.