On a variational problem of nematic liquid crystal droplets

TL;DR

This paper proves the existence and uniqueness of minimizers for a variational problem modeling nematic liquid crystal droplets, involving energy functionals with domain and boundary conditions, among specific classes of domains.

Contribution

It establishes the existence and uniqueness of minimizers for a class of energy functionals related to nematic liquid crystals, considering domain regularity and boundary conditions.

Findings

Existence of minimizers for the energy functional with fixed volume domains.

Uniqueness of optimal configurations in various settings.

Existence of minimizers for a generalized energy functional with boundary-dependent terms.

Abstract

Let $\mu>0$ be a fixed constant, and we prove that minimizers to the following energy functional \begin{align*} E_f(u,\Omega):=\int_{\Omega}|\nabla u|^2+\mu P(\Omega) \end{align*}exist among pairs $(\Omega,u)$ such that $\Omega$ is an $M$-uniform domain with finite perimeter and fixed volume, and $u \in H^1(\Omega,\mathbb{S}^2)$ with $u =\nu_{\Omega}$, the measure-theoretical outer unit normal, almost everywhere on the reduced boundary of $\Omega$. The uniqueness of optimal configurations in various settings is also obtained. In addition, we consider a general energy functional given by \begin{align*} E_f(u,\Omega):=\int_{\Omega} |\nabla u(x)|^2 \,dx + \int_{\partial^* \Omega} f\big(u(x)\cdot \nu_{\Omega}(x)\big) \,d\mathcal{H}^2(x), \end{align*}where $\partial^* \Omega$ is the reduced boundary of $\Omega$ and $f$ is a convex positive function on $\mathbb R$. We prove that minimizers of $E_f$ also exist among $M$-uniform outer-minimizing domains $\Omega$ with fixed volume and $u \in H^1(\Omega,\mathbb{S}^2)$.