On a variational problem of nematic liquid crystal droplets

Abstract

Let $\mu >0$ be a fixed constant, and we prove that minimizers to the following energy functional \begin{align*} E_f(u,Ω):=\int_Ω|\nabla u|^2+μP(Ω) \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,Ω):=\int_Ω |\nabla u(x)|^2 \,dx + \int_{\partial^* Ω} f\big(u(x)\cdot ν_Ω(x)\big) \,d\mathcal{H}^2(x), \end{align*}where ${\partial }^{\ast }\Omega$ 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 $u\in H^1(\Omega ,{\mathbb{S}}^2)$.

19 pages