Boundary regularity of stationary critical points for a Cosserat energy functional
Tóm tắt
In this paper, we will discuss the boundary regularity of stationary critical points for the following Cosserat energy functional:
0.1
$$\begin{aligned} {\textrm{Coss}}(\phi , R)=\int _\Omega \big (|R^t\nabla \phi -I_3|^2+|\nabla R|^p+\langle \phi -x, f\rangle +\langle R, M\rangle \big )\,dx, \end{aligned}$$
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
$$\Omega _{\delta }:=\{x\in {\bar{\Omega }}, dist(x,\partial \Omega )\le \delta \}$$
,
$$\delta >0$$
.
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