A representation formula for members of SBV dual

TL;DR

This paper introduces an integral representation formula for the dual space of SBV functions, utilizing a new fiber space structure that captures local geometric information of rectifiable sets.

Contribution

It provides a novel integral representation for SBV dual elements using a specially constructed fiber space with approximate germs of rectifiable sets.

Findings

New fiber space structure for SBV duals

Integral representation formula for dual elements

Framework for analyzing SBV dual spaces

Abstract

We give an integral representation formula for members of the dual of $SBV(\mathbb{R}^n)$ in terms of functions that are defined on $\hat{\mathbb{R}}^n$, an appropriate fiber space that we introduce, consisting of pairs $(x,[E]_x)$ where $[E]_x$ is an approximate germ of an $(n-1)$-rectifiable set $E$ at $x$.