Sphere-valued harmonic maps with surface energy and the $K_{13}$ problem

TL;DR

This paper investigates a sphere-valued energy functional inspired by the $K_{13}$ problem in nematic liquid crystals, focusing on conditions for boundedness, critical points, and the partial regularity of minimizers.

Contribution

It introduces a new energy functional with a surface term, analyzes conditions for boundedness, and studies the regularity of its minimizers under specific boundary conditions.

Findings

Energy is unbounded from below without boundary restrictions.

Suitable boundary conditions make the energy bounded.

Partial regularity results for minimizers are established.

Abstract

We consider an energy functional motivated by the celebrated $K_{13}$ problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimizers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the minimizers.