论文标题
恒定卷曲的旋转是恒定的
Rotations with Constant Curl are Constant
论文作者
论文摘要
经典的结果是，如果$ u \ in c^2（\ mathbb {r}^n; \ mathbb {r}^n）$和$ \ nabla u \在so（n）$中，则遵循$ u $是固定的。在本文中，该结果概括为具有非散曲线的矩阵字段。结果表明，每个矩阵字段$ r \ in c^2（ω\ subseteq \ mathbb {r}^3; so（3））$，使得$ \ operatatorName {curl} r =常数$必然是常数。此外，在任意维度上证明了可测量的旋转场与其分布卷曲所允许一样规律。特别是，一个可测量的矩阵字段$ r：ω\ to so（n）$，其在分布意义上的卷发也很光滑。
It is a classical result that if $u \in C^2(\mathbb{R}^n;\mathbb{R}^n)$ and $\nabla u \in SO(n)$ it follows that $u$ is rigid. In this article this result is generalized to matrix fields with non-vanishing curl. It is shown that every matrix field $R\in C^2(Ω\subseteq \mathbb{R}^3;SO(3))$ such that $\operatorname{curl } R = constant$ is necessarily constant. Moreover, it is proved in arbitrary dimensions that a measurable rotation field is as regular as its distributional curl allows. In particular, a measurable matrix field $R: Ω\to SO(n)$, whose curl in the sense of distributions is smooth, is also smooth.
