We use the method of Γ-convergence to study the behavior of the Landau-1 de Gennes model for a nematic liquid crystalline film in the limit of vanishing thickness. In this asymptotic regime, surface energy plays a greater role and we take particular care in understanding its influence on the structure of the minimizers of the derived two-dimensional energy. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and the strong Dirichlet boundary conditions on the lateral boundary of the film. The constants in the weak anchoring conditions are chosen so as to enforce that a surface-energy-minimizing nematic Q-tensor has the normal to the film as one of its eigenvectors. We establish a general convergence result and then discuss the limiting problem in several parameter regimes.
We study tensor-valued minimizers of the Landau-de Gennes energy functional on a simply-connected planar domain Ω with noncontractible boundary data. Here the tensorial field represents the second moment of a local orientational distribution of rod-like molecules of a nematic liquid crystal. Under the assumption that the energy depends on a single parameter-a dimensionless elastic constant ε > 0-we establish that, as ε → 0, the minimizers converge to a projectionvalued map that minimizes the Dirichlet integral away from a single point in Ω. We also provide a description of the limiting map.
We consider the three-dimensional Ginzburg-Landau model for a solid spherical superconductor in a uniform magnetic field, in the limit as the Ginzburg-Landau parameter κ = 1/ε → ∞. By studying a limiting functional we identify a candidate for the lower critical field H c 1 , the value of the applied field strength at which minimizers first exhibit vortices. For applied fields of this strength we show the existence of locally minimizing solutions with vortices located along a diameter of the sphere parallel to the applied field direction. To analyze these problems we use a combination of techniques, involving least perimeter problems, weak Jacobians and rectifiable currents, and special Hodge decompositions. 2005 Elsevier SAS. All rights reserved.
RésuméNous étudions la limite quand le paramètre de Ginzburg-Landau κ = 1/ε → ∞ pour le modèle de Ginzburg-Landau en trois dimension dans le cas d'une boule placée dans un champ magnétique uniforme. Nous identifions une fonctionnelle limite qui nous permet de trouver le premier champ critique H c 1 , c'est à dire le champ au dessus duquel les minimiseurs commencent à presenter des vortex. Nous montrons qu'il existe des solutions localement minimisantes ayant des vortex le long du diamètre de la boule qui est parallèle au champ appliqué quand sa norme est de l'ordre de H c 1 . Nous nous servons de techniques provenant de la théorie de la mesure géométrique, incluant les jacobiens faibles et les courants rectifiables, ainsi que de techniques provenant de problèmes de minimisation de périmètre.
We use the method of Γ-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach in [1] where we considered a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible 1
We construct local minimizers to the Ginzburg-Landau energy in certain threedimensional domains based on the asymptotic connection between the energy and the total length of vortices using the theory of weak Jacobians. Whenever there exists a collection of locally minimal line segments spanning the domain, we can find local minimizers with arbitrarily assigned degrees with respect to each segment.
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