# A Friedrichs-Maz'ya inequality for functions of bounded variation

## TL;DR

This paper provides an elementary, general proof of Maz'ya's inequality for functions of bounded variation, extending and optimizing previous versions to arbitrary domains with sharp conditions.

## Contribution

It introduces a new, elementary proof method that extends Maz'ya's inequality to all domains, including those with complex boundaries, and establishes sharp conditions for bounded variation extension.

## Key findings

- The inequality is optimal in several respects.
- Provides necessary and sufficient conditions for BV extension on domains with $\sigma$-finite boundary.
- Shows the general sufficient condition is sharp via counterexample.

## Abstract

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that it is rather elementary, if the basic background in the theory of functions of bounded variation is assumed. Never the less, it allows to extend all the previously known versions of this fundamental inequality to a completely general version. In fact the inequality presented here is optimal in several respects. As already observed in previous proofs, the crucial step is to provide conditions under which a function of bounded variation on a bounded open set, extended to zero outside, has bounded variation on the whole space. We push such conditions to their limits. In fact, we give a sufficient and necessary condition if the open set has a boundary with $\sigma$-finite surface measure and a sufficient condition if the open set is fully arbitrary. Via a counterexample we show that such a general sufficient condition is sharp.