Improved partial regularity for manifold-constrained minimisers of subquadratic energies

TL;DR

This paper studies the regularity of energy-minimizing maps from 3D domains to the real projective plane, revealing a detailed structure of their singularities relevant to nematic liquid crystals.

Contribution

It provides a refined partial regularity result for manifold-constrained minimizers of subquadratic energies, identifying the structure of singular sets in nematic liquid crystal models.

Findings

Singular set decomposes into a 1D non-orientable defect line and finite point defects.

Maps minimize p-harmonic energy for 1<p<2, relevant to nematic liquid crystals.

Enhanced understanding of defect structures in liquid crystal models.

Abstract

We consider minimising $p$-harmonic maps from three-dimensional domains to the real projective plane, for $1<p<2$. These maps arise as least-energy configurations in variational models for nematic liquid crystals. We show that the singular set of such a map decomposes into a $1$-dimensional set, which can be physically interpreted as a non-orientable line defect, and a locally finite set, i.e. a collection of point defects.