# MLC

## Seminar

### Analysis of defects in minimizers for a planar Frank energy

Satellite

#### Abstract

Smectic C* liquid crystal films are modeled with a relaxed Frank energy, \begin{equation*} \int_\Omega\Big( k_s(\text{div}\, u)^2 + k_b(\text{curl}\, u)^2 + \frac{1}{2\epsilon^2}(1 - |u|^2)^2 \Big)\, dx . \end{equation*} Here $k_s$ and $k_b$ represent the two dimensional splay and bend moduli for the film respectively with $k_s, k_b > 0$, $\Omega$ is a planar domain, and $u$ is an $\mathbb{R}^2$-valued vector field with fixed boundary data having degree $d>0$. We study the limiting pattern for a sequence of minimizers $\{u_\epsilon\}$ as $\epsilon\to 0$. We prove that the pattern contains $d$, degree one defects and that it has a either a radial or circular asymptotic form near each defect depending on the relative values of $k_s$ and $k_b$. We further characterize a renormalized energy for the problem and show that it is minimized by the limit. This is joint work with Sean Colbert-Kelly.#### Video

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