Trace of BV-functions on irregular subsets

TL;DR

This paper extends the theory of boundary traces of BV-functions to more general irregular sets, broadening the applicability beyond regions with finite perimeter and almost everywhere existing normals.

Contribution

It generalizes boundary trace results of BV-functions from regions with finite perimeter to arbitrary countably rectifiable sets, removing the need for normals to exist almost everywhere.

Findings

Boundary trace of BV-functions defined on countably rectifiable sets.

Generalization of Maz'ya's results to irregular subsets.

Broader applicability of BV-function boundary analysis.

Abstract

In the paper, the basic results on boundary trace of the book "Sobolev spaces" by V. Maz'ya are generalized to a wider class of regions. In the book, boundary trace of BV-functions is defined for regions 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 of that, we investigate trace of BV functions on any countably (n-1)-rectifiable set.