# Energy Minimization Problem related to certain Liquid Crystal Models with respect to the Position, Orientation and Size of an Obstacle having Dihedral Symmetry

## TL;DR

This paper studies an energy minimization problem involving obstacle placement with dihedral symmetry inside a disk, relevant to liquid crystal models, analyzing optimal configurations and sensitivities of the system.

## Contribution

It introduces a novel framework for optimizing obstacle placement with dihedral symmetry in liquid crystal models, considering translations, rotations, and size variations.

## Key findings

- Identifies extremal obstacle configurations minimizing energy.
- Analyzes sensitivity of energy to boundary data and obstacle parameters.
- Provides conditions under which symmetric obstacles optimize the energy.

## Abstract

In this paper, we deal with an obstacle placement problem inside a disk, that can be formulated as an energy minimization problem with respect to the rotations of the obstacle about its center, with respect to the translations of the obstacle within a planar disk and also with respect to the expansion or contraction of the obstacle inside the disk. We also include its sensitivity with respect to the boundary data. We consider an inhomogeneous Dirichlet Boundary Value Problem on $ B \setminus P$ as follows: $\Delta u =0$ on $ B \setminus P$, $u=0$ on $\partial P$, $u=M$, a nonzero constant on $\partial B$. Here, the planar obstacle $P$ is invariant under the action of a dihedral group $\mathbb{D}_n$, $n \in \mathbb{N}$, $n \geq 3$, and $B$ is a planar disk containing $P$. We make the following assumptions on $P$ and $B$: (0) the volume constraint on $P$ and $B$ both, (i) invariance of both $B$ and $P$ under the action of the same dihedral group, (ii) $B$ and $P$ need not be concentric, (iii) smoothness condition on both the boundaries, (iv) a monotonicity condition on the boundary $\partial P$ of the obstacle $P$. Examples of such obstacles include regular polygons with smoothened corners, ellipses, among other star-shaped domains. In this setting, we investigate the extremal configurations of the obstacle $P$ with respect to the disk $B$, for the energy functional $E$ associated with the boundary value problem. The Dirichlet energy is the one-constant Oseen-Frank energy of a uniaxial planar nematic liquid crystal. The class of problems considered here sheds some light into optimal locations of holes with a certain symmetry inside a nematic-filled disc, so as to optimise the corresponding one-constant nematic Oseen-Frank energy.