经典的结果是，如果$u \in C^2(\mathbb{R}^n;\mathbb{R}^n)$和$\nabla u \in SO(n)$，则$u$是刚性的。本文将这一结果推广到具有非零旋度的矩阵场。证明了每个矩阵域$R\in C^2(Ω\subseteq \mathbb{R}^3;SO(3))$使得$\operatorname{curl } R = constant$必为常数.此外，还证明了在任意维数下，可测旋转场在其分布旋度允许的范围内是正则的。特别是，一个可测矩阵场$R: Ω\to SO(n)$，其分布意义上的旋度是光滑的，也是光滑的。
英文摘要：
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.
Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients
考虑了$\mathbb{R}^3$中的非线性卷曲-卷曲问题$\nabla\times\nabla\times U + V(x) U=f(x,|U|^2)U$，该问题与具有Kerr-type非线性物质定律的非线性Maxwell方程组有关.我们证明了一个有界的，圆柱对称系数$V$和亚临界圆柱对称非线性$f$的对称基态解的存在性。新的存在性结果扩展了一类问题的基态类型的解决方案是已知的。它是基于紧性的对称函数由于狮子，新的重排型不等式布洛克和最近的扩展Nehari流形技术由Szulkin和Weth。
The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates
Hanoi University of Science and Technology（河内科技大学）；FPT University（FPT大学）
