Publication 20-CNA-014

Rotations with Constant Curl are Constant

Janusz Ginster
Institut für Mathematik
Humboldt-Universität zu Berlin
12489 Berlin
janusz.ginster@math.hu-berlin.de

Amit Acharya
Dept. of Civil & Environmental Engineering
Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh, PA 15213
acharyaamit@cmu.edu

Abstract: We address a problem that extends a fundamental classical result of continuum mechanics from the time of its inception, as well as answers a fundamental question in the recent, modern nonlinear elastic theory of dislocations. Interestingly, the implication of our result in the latter case is qualitatively different from its well-established analog in the linear elastic theory of dislocations.

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 nonvanishing curl. It is shown that every matrix field $R \in {C}^2 ( \Omega \subseteq \mathbb{R}^3; SO(3))$ such that 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 $R : \Omega \rightarrow SO(n)$, whose curl in the sense of distributions is smooth, is also smooth.

Get the paper in its entirety as  20-CNA-014.pdf

---

«   Back to CNA Publications