Related papers: Liquid crystals and harmonic maps in polyhedral do…
We consider a non-local free energy functional, modelling a competition between entropy and pairwise interactions reminiscent of the second order virial expansion, with applications to nematic liquid crystals as a particular case. We build…
We model nematic liquid crystal configurations inside three-dimensional prisms, with a polygonal cross-section and Dirichlet boundary conditions on all prism surfaces. We work in a reduced Landau-de Gennes framework, and the Dirichlet…
We study nematic liquid crystal configurations in a prototype bistable device - the Post Aligned Bistable Nematic (PABN) cell. Working within the Oseen-Frank continuum model, we describe the liquid crystal configuration by a unit-vector…
We summarise some recent results on solution landscapes for two-dimensional (2D) problems in the Landau--de Gennes theory for nematic liquid crystals. We study energy-minimizing and non energy-minimizing solutions of the Euler--Lagrange…
The phase-field-crystal model for liquid crystals is solved numerically in two spatial dimensions. This model is formulated with three position-dependent order parameters, namely the reduced translational density, the local nematic order…
We propose a model for nonlinearly elastic membranes undergoing finite deformations while confined to a regular frictionless surface in $\mathbb{R}^3$. This is a physically correct model of the analogy sometimes given to motivate harmonic…
Given a half-harmonic map $u\in \dot H^{\frac{1}{2},2}(\mathbb{R},\mathbb{S}^1)$ minimizing the fractional Dirichlet energy under Dirichlet boundary conditions in $\mathbb{R}\setminus I$, we show the existence of a second half-harmonic map,…
We generalize the shape optimization problem for the existence of stable equilibrium configurations of nematic and cholesteric liquid crystal drops surrounded by an isotropic solution to include a broader family of admissible domains with…
The Hodge equations for 1-forms are studied on Beltrami's projective disc model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for…
The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled…
We consider a class of liquid crystal free-boundary problems for which both the equilibrium shape and internal configuration of a system must simultaneously be determined, for example in films with air- or fluid-liquid crystal interfaces…
We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-\Delta u=\frac{\lambda f(x)}{(1-u)^2}$ on a bounded domain $\Omega \subset \R^N$, which…
Within the Landau-de Gennes theory of liquid crystals, we study theoretically the equilibrium configurations with uniaxial symmetry. For an arbitrary form of the bulk energy density, we show that energy minimizers among uniaxially symmetric…
This paper outlines an energy-minimization finite-element approach to the modeling of equilibrium configurations for nematic liquid crystals in the presence of internal and external electric fields. The method targets minimization of system…
The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields $H^1(S,T)$, where $S$ and $T$ are surfaces of revolution. The energy functional we consider is closely related…
This paper is concerned with the rigorous analysis of a recently proposed model of Zheng et. al. for describing nematic liquid crystals within the dense regime, with the orientation distribution function as the variable. A key feature of…
Given two-dimensional Riemannian manifolds $\mathcal{M},\mathcal{N}$, we prove a lower bound on the distortion of embeddings $\mathcal{M} \to \mathcal{N}$, in terms of the areas' discrepancy $V_{\mathcal{N}}/V_{\mathcal{M}}$, for a certain…
We develop a novel finite element method for a phase field model of nematic liquid crystal droplets. The continuous model considers a free energy comprised of three components: the Ericksen's energy for liquid crystals, the Cahn-Hilliard…
We show that the standard Fermi--Pasta--Ulam system, with a suitable choice for the interparticle potential, constitutes a model for glasses, and indeed an extremely simple and manageable one. Indeed, it allows one to describe the landscape…
A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This…