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arXiv · 2609.36972

On the minimality of the radial singularity in nematic liquid crystals

Pierre Bousquet·Xavier Lamy
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Abstract

We show that the map $u_*\colon B_1\subset\mathbb R^3\to\mathbb S^2$, $x\mapsto x/|x|$, 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 $k_1\leq k_2$. Our main tool is a new stability estimate for $u_*$ as a minimizing sphere-valued harmonic map.

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Pierre Bousquet, Xavier Lamy. 2026-09-29. On the minimality of the radial singularity in nematic liquid crystals. https://arxiv.org/abs/2609.36972

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