Approximation of BV by SBV functions in metric spaces

TL;DR

This paper demonstrates that in certain metric spaces, BV functions can be approximated uniformly and strictly by special BV functions, using variational capacity as a key tool.

Contribution

It introduces a method to approximate BV functions by SBV functions in metric spaces with doubling measures and Poincaré inequalities, without increasing jump discontinuities.

Findings

BV functions can be approximated by SBV functions in the specified metric spaces.

The variational 1-capacity and its BV analog are effective tools for approximation.

Approximation preserves key properties without adding significant jumps.

Abstract

In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we show that functions of bounded variation (BV functions) can be approximated in the strict sense and pointwise uniformly by special functions of bounded variation, without adding significant jumps. As a main tool, we study the variational 1-capacity and its BV analog.