论文标题
梯度矢量场的规律性产生有限的caccioppoli分区
Regularity of gradient vector fields giving rise to finite Caccioppoli partitions
论文作者
论文摘要
对于有限集$ a \ subseteq \ mathbb {r}^n $，请考虑一个函数$ u \ in \ mathrm {bv} _ {\ mathrm {\ mathrm {loc}}}^2（\ mathbb {r}^n）$，这样几乎在$几乎在$中，如果$ a $是独立的，则遵循$ u $是分段仿射，远离封闭的$ \ mathcal {h}^{n -1} $ - 可纠正的设置。如果$ a $是独立的，则$ u $是分段仿射，远离封闭的$ \ mathcal {h}^{n -1} $ - null set。
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.
