We prove full boundary regularity for minimizers \(f=(\varphi ,R)\in H^1(\Omega ,\mathbb {R}^3)\times W^{1,p}(\Omega ,SO(3))\) of the Cosserat energy functional \(\begin{aligned} \mathcal {J}(\varphi ,R) = \mathop {\int }\nolimits _{\Omega } |R^T\textrm{D}\varphi -I_3{|}^2 + |\textrm{D}R{|}^p \textrm{d}{x} \end{aligned}\) with respect to \(C^1-\) Dirichlet boundary data, when \(p\in [2,3]\) . Contrary to existing results, we need not assume that the minimizer fulfils a boundary monotonicity formula, because we prove such a formula for minimizers. In particular, this partially improves current results by Li and Wang, who showed partial boundary regularity for stationary critical points of the above Cosserat energy.
