On the minimality of the radial singularity in nematic liquid crystals

We show that the map $u_{\ast }\text{\hspace{0.17em}⁣}:B_1\subset {\mathbb{R}}^3\to {\mathbb{S}}^2$, $x\mapsto x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$, minimizes the anisotropic energy \begin{align*} \int_{B_1} \Big( k_1($\mathrm{d}\mathrm{i}\mathrm{v}$, u)^2+k_2 (u\cdot $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}$, u)^2 +k_3 |u\times$\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}$, 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.