Boundary regularity of stationary critical points for a Cosserat energy functional

Abstract

In this paper, we will discuss the boundary regularity of stationary critical points for the following Cosserat energy functional:

$$\begin{aligned} {\textrm{Coss}}(\phi , R)=\int _\Omega \big (|R^t\nabla \phi -I_3|^2+|\nabla R|^p+\langle \phi -x, f\rangle +\langle R, M\rangle \big )\,dx, \end{aligned}$$

(0.1)

where $2\le p<3$, $f\in L^{\infty }(\Omega , {\mathbb {R}}^3)$, $M\in L^{\infty }(\Omega , SO(3))$, and $\Omega \subset \mathbb {R}^3$ is a domain with $C^1$ boundary. Precisely, if $(\phi , R)\in H^1(\Omega ,{\mathbb {R}}^3)\times W^{1,p}(\Omega , SO(3))$ is a stationary critical point of (0.1) satisfying a certain boundary monotonicity inequality, we show that there exists a closed subset $\Sigma \subset \partial \Omega$ satisfying the Hausdorff measure $H^{3-p}(\Sigma )=0$ such that $(\phi , R)\in C^{1,\alpha }(\Omega _{\delta }\setminus \Sigma )\times C^\alpha (\Omega _{\delta }\setminus \Sigma )$, where $\Omega _{\delta }:=\{x\in {\bar{\Omega }}, dist(x,\partial \Omega )\le \delta \}$, $\delta >0$.