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Let $\mu>0$ be a fixed constant, and we prove that minimizers to thefollowing energy functional \begin{align*}  E_f(u,\Omega):=\int_{\Omega}|\nabla u|^2+\mu P(\Omega) \end{align*}existamong pairs $(\Omega,u)$ such that $\Omega$ is an $M$-uniform domain withfinite 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 onthe reduced boundary of $\Omega$. The uniqueness of optimal configurations invarious settings is also obtained. In addition, we consider a general energyfunctional 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 convexpositive function on $\mathbb R$. We prove that minimizers of $E_f$ also existamong $M$-uniform outer-minimizing domains $\Omega$ with fixed volume and $u\in H^1(\Omega,\mathbb{S}^2)$.

On a variational problem of nematic liquid crystal droplets