ABSTRACT

In this talk, we will describe an energy minimizing problem arising from seeking the optimal configurations of a class of nematic liquid crystal droplets. More precisely, the general problem seeks a pair $(\Omega, u)$ that minimizes the energy functional: $$E(u,\Omega)= \int_\Omega \frac12|\nabla u|^2+ \mu \int_{\partial\Omega} f(x,u(x)) d\sigma,$$ among all open set $\Omega$ within the unit ball of $\mathbb R^3$ , with a fixed volume, and $u\in H^1(\Omega,\mathbb S^2)$. Here $f:\mathbb R^3\times \mathbb R \to\mathbb R$ is a suitable nonnegative function, which is given. While the existence of minimizers remains open in the full generality, there has been some partial progress when $\Omega$ is assumed to be convex. In this talk, I will discuss some results for $\Omega$ that are not necessarily convex. This is a joint work with my student Qinfeng Li.