Abstract:

This paper investigates a Liouville type theorem for three-dimensional stationary liquid crystal equations. We show that if $u\in{L^6}({\mathbb{R}^3}) \cap {L^ q}({\mathbb{R}^3})$, $\nabla d\in{L^2}({\mathbb{R}^3}) \cap {L^q}({\mathbb{R}^3})$ and the anisotropic integrability conditions of ${u_i} \in L_{{x_i}}^{\frac{q}{{q - 2}}}L_{{{\tilde x}_i}}^s(\mathbb{R} \times {\mathbb{R}^2}), \forall i = 1,2,3$, $\frac{2}{q} + \frac{1}{s} \ge \frac{1}{2}$, $2 < q < \infty,1 \le s \le \infty$ are satisfied, then $u=0, \nabla d = 0$.

Key words: Nematic liquid crystal equations, The Liouville problem, Anisotropy