We investigate prototypical profiles of point defects in two dimensional liquid crystals within the framework of Landau-de Gennes theory. Using boundary conditions characteristic of defects of index k/2, we find a critical point of the Landau-de Gennes energy that is characterised by a system of ordinary differential equations. In the deep nematic regime, b 2 small, we prove that this critical point is the unique global minimiser of the Landau-de Gennes energy. For the case b 2 = 0, we investigate in greater detail the regime of vanishing elastic constant L → 0, where we obtain three explicit point defect profiles, including the global minimiser.
Nowadays, nonhomogeneous and periodic ferromagnetic materials are the subject of a growing interest. Actually such periodic configurations often combine the attributes of the constituent materials, while sometimes, their properties can be strikingly different from the properties of the different constituents. These periodic configurations can be therefore used to achieve physical and chemical properties difficult to achieve with homogeneous materials. To predict the magnetic behavior of such composite materials is of prime importance for applications.The main objective of this paper is to perform, by means of Γ-convergence and two-scale convergence, a rigorous derivation of the homogenized Gibbs-Landau free energy functional associated to a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the ferromagnetic media. We thus describe the Γ-limit of the Gibbs-Landau free energy functional, as the period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to zero.
The main aim of this note is to prove a sharp Poincaré-type inequality for vectorvalued functions on S 2 , that naturally emerges in the context of micromagnetics of spherical thin films.
The paper is about the parking 3-sphere swimmer ($\text{sPr}_3$). This is a low-Reynolds number model swimmer composed of three balls of equal radii. The three balls can move along three horizontal axes (supported in the same plane) that mutually meet at the center of $\text{sPr}_3$ with angles of $120^{\circ}$. The governing dynamical system is introduced and the implications of its geometric symmetries revealed. It is then shown that, in the first order range of small strokes, optimal periodic strokes are ellipses embedded in 3d space, i.e., closed curves of the form $t\in [0,2\pi] \mapsto (\cos t)u + (\sin t)v$ for suitable orthogonal vectors $u$ and $v$ of $\mathbb{R}^3$. A simple analytic expression for the vectors $u$ and $v$ is derived. The results of the paper are used in a second article where the real physical dynamics of $\text{sPr}_3$ is analyzed in the asymptotic range of very long arms.
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