Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

Abstract

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