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

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 ${\mathbb{D}}_n$, $n\in \mathbb{N}$, $n\ge 3$, 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.

17 pages, 8 figures, 7 tables. arXiv admin note: text overlap with arXiv:1707.01368