Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

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.