The Lipschitz truncation of functions of bounded variation

TL;DR

This paper introduces a Lipschitz truncation method for functions of bounded variation that approximates them in the area-strict metric while only altering the original function on a small set, improving upon previous measure-based estimates.

Contribution

It develops a new Lipschitz truncation technique for BV functions that works in the area-strict metric, with minimal modifications to the original function.

Findings

Lipschitz truncation approximates BV functions in the area-strict metric.

The truncation alters the original function only on a small set.

Previous estimates were limited to measure of the difference set.

Abstract

We construct a Lipschitz truncation which approximates functions of bounded variation in the area-strict metric. The Lipschitz truncation changes the original function only on a small set similar to Lusin's theorem. Previous results could only give estimates on the Lebesgue measure of the set where the Lipschitz approximations differ from the original function.