Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

TL;DR

This paper investigates the regularity of gradient vector fields associated with finite sets in Euclidean space, showing they induce piecewise affine functions with specific regularity properties depending on the independence conditions of the set.

Contribution

It establishes conditions under which functions with gradients in a finite set are piecewise affine, extending understanding of their regularity based on convex and affine independence.

Findings

Functions with gradients in a finite set are piecewise affine outside a rectifiable set.

Convex independence of the set implies piecewise affinity away from a rectifiable set.

Affine independence leads to piecewise affinity outside a null set.

Abstract

For a finite set $A \subseteq \mathbb{R}^n$, consider a function $u \in \mathrm{BV}_{\mathrm{loc}}^2(\mathbb{R}^n)$ such that $\nabla u \in A$ almost everywhere. If $A$ is convex independent, then it follows that $u$ is piecewise affine away from a closed, countably $\mathcal{H}^{n - 1}$-rectifiable set. If $A$ is affinely independent, then $u$ is piecewise affine away from a closed $\mathcal{H}^{n - 1}$-null set.