On the minimality of the radial singularity in nematic liquid crystals

Abstract

We show that the map $u_{\ast }:B_1\subset {\mathbb{R}}^3\to {\mathbb{S}}^2$, $x\mapsto x/|x|$, minimizes the anisotropic energy \begin{align*} \int_{B_1} \Big( k_1(\mathrm{div}\, u)^2+k_2 (u\cdot \mathrm{curl}\, u)^2 +k_3 |u\times\mathrm{curl}\, u|^2 \Big) \, dx, \end{align*} among ${\mathbb{S}}^2$-valued maps agreeing with it on the boundary, for values of $k_1,k_2,k_3>0$ beyond the known regime $k_1\le k_2$. Our main tool is a new stability estimate for $u_{\ast }$ as a minimizing sphere-valued harmonic map.