Rotations with Constant $$\mathbf {{\text {curl }}}$$ are Constant

Abstract

AbstractWe 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 $$u\in C^2({\mathbb {R}}^n;{\mathbb {R}}^n)$$ u ∈ C 2 ( R n ; R n ) and $$\nabla u \in SO(n)$$ ∇ u ∈ S O ( n ) , it follows that u is rigid. In this article this result is generalized to matrix fields with non-vanishing $${\text {curl }}$$ curl . It is shown that every matrix field $$R\in C^2(\varOmega ;SO(3))$$ R ∈ C 2 ( Ω ; S O ( 3 ) ) such that $${\text {curl }}R = constant$$ curl R = c o n s t a n t is necessarily constant. Moreover, it is proved in arbitrary dimensions that a measurable rotation field is as regular as its distributional $${\text {curl }}$$ curl allows. In particular, a measurable matrix field $$R: \varOmega \rightarrow SO(n)$$ R : Ω → S O ( n ) , whose $${\text {curl }}$$ curl in the sense of distributions is smooth, is also smooth.