Trace inequality for $BV(\Omega)$ in smooth domains

TL;DR

This paper establishes an optimal trace inequality for functions of bounded variation in smooth domains, highlighting the importance of domain regularity for the inequality's validity.

Contribution

It provides the sharpest version of the trace inequality for BV functions, emphasizing the domain regularity conditions needed.

Findings

Optimal trace inequality for BV functions proved

Regularity conditions on the domain identified as crucial

Enhanced understanding of boundary behavior of BV functions

Abstract

This work is devoted to prove an optimum version of the trace inequality associated to the embedding $BV(\Omega)\subset L^1(\partial\Omega)$. Special emphasis is placed on the regularity that the domain $\Omega$ should exhibit for this result to be valid.