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.

References in corpus (1)

• Dynamics of Phase Separation of Crystal Surfaces