Radial Symmetry on 3D Shells in the Landau-de Gennes Theory

TL;DR

This paper analyzes the stability and minimality of the radial-hedgehog solution in 3D spherical shells within the Landau-de Gennes theory, establishing conditions for its uniqueness and global minimality across temperature regimes.

Contribution

It provides the first rigorous proof of the radial-hedgehog solution's stability and global minimality in 3D shells, including explicit geometric criteria and low-temperature results.

Findings

Radial-hedgehog solution has no zeroes in sufficiently narrow shells below nematic supercooling temperature.

It is the unique global energy minimizer in such shells.

Global minimality holds at low temperatures for all inner and outer radii.

Abstract

We study the stability of the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. We show that the radial-hedgehog solution has no zeroes for a sufficiently narrow shell, for all temperatures below the nematic supercooling temperature. We prove that the radial-hedgehog solution is the unique global Landau-de Gennes energy minimizer for a sufficiently narrow 3D spherical shell, for all temperatures below the nematic supercooling temperature. We provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution in this temperature regime. In the low temperature limit, we prove the global minimality of the radial-hedgehog solution on a 3D spherical shell, for all values of the inner and outer radii.