We study global minimizers of the Landau-de Gennes energy for the nematic liquid crystals in simply connected, bounded, smooth domains of dimension 3, subject to a weak planar anchoring. The boundary condition is degenerate. In the regime where the elastic constant tends to zero and the temperature is below the nematic-isotropic transition threshold, the bulk and surface energy enforce the energy minimizers to be uniaxial, with director fields lying in the tangent plane to the boundary in the sense of trace. We establish that the singular set of the limiting minimizer within the closure of the domain is a finite set by studying the associated harmonic map with a strong tangential boundary constraint. The structure of the singularities located on the boundary is also investigated.
in 1987. In certain parameter regimes, the hedgehog solution is unstable. More defect structures may occur to stabilize the critical points of the Landau-de Gennes energy. For example, the half-degree biaxial ring and the split-core segment of strength 1 are two topologically different disclinations. They were identified through numerical simulations by Penzenstadler-Trebin [34] in 1989 and Gartland-Mkaddem [33] in 2000, respectively. Recently, mathematical proofs for the existence of ring and split-core disclinations in the Landau-de Gennes theory have been rigorously shown by Yu in [40] for the limit case and by Tai-Yu in [38] for the low-temperature case. We also refer readers to the series of works by Dipasquale-Millot-Pisante for the Landau-de Gennes theory with Lyuksyutov constraint. See [14,15,16]. More studies on the defect structures include the works by Canevari [9], Bauman-Park-Phillips [5], Di Fratta-Robbins-Slastikov-Zarnescu [13], and Ignat-Nguyen-Slastikov-Zarnescu [26,27,28].
In the previously mentioned articles, the defects are located in the interior of the bulk domain. In practice, the defects on the bounding interface are also of significant interest for both fundamental research and technological applications. A boojum singularity is a topologically stable point defect that forms on the bounding interface. To study the boojums, we need appropriate boundary conditions for the director fields. In general, the alignment of the director fields on the bounding interface depends on the anchoring effect, which can be prescribed in a strong or weak sense. Strong anchoring is given by the Dirichlet conditions imposed on the interface. Weak anchoring can be realized by penalizing the free energy with a surface integral. Usually, weak anchoring conditions are more realistic than the strong ones. When the weak anchoring is degenerate, it is believed that the boojum singularities may occur. This article focuses on the Landau-de Gennes theory with degenerate weak planar anchoring. The bulk is a simply connected, bounded, smooth domain of dimension 3. Particular attention will be paid to the limit case while the elastic constant approaches 0. Our motivations mainly originate from the boojum singularities observed in various physics literature. See Volovik-Lavrentovich [39], Lavrentovich [29] and Fournier-Galatola [20].
Before introducing our degenerate weak planar anchoring, let us first discuss some existing works related to the non-degenerate weak anchoring. In this case, the surface integral in the total energy measures the L 2 -distance between the order parameter and a given map defined on the boundary. It is non-degenerate in the sense that its director field favors some particular directions prescribed by the given map. Contreras-Lamy-Rodiac [10] studied the Oseen-Frank model in one-constant approximation. In 3D, there is no defect on the boundary of the domain. Huang-Wang [25] investigated the bubbling phenomena of the approximate weak harmonic maps in 2D, up to the boundary. Alama-Bronsard-Galvão-Sousa [1] studied the Landau-de Gennes model with non-degenerate weak anchoring in 2D domains. The original problem is reduced to the Ginzburg-Landau model with a complex-valued order parameter. As the length scale parameter ϵ tends to 0 and when anchoring strength converges at a specific rate to ϵ, vortices can appear on the boundary of the domain. The topological degrees of the vortices are +1 for bounded simply connected domains, and are -1 for annulus domains or exterior domains. These vortices can be interpreted as cross-sections of disclination lines in 3D domains. Bauman-Phillips-Wang [6] examined a similar problem in higherdimensional domains. They showed that the minimizers converge to the generalized harmonic map smoothly in the interior, away from a (n -2)-rectifiable set of finite (n -2)-Hausdorff measure.
Under a suitable η-smallness condition, they eliminated the possibility of forming vortices on the boundary. Moreover, when the anchoring strength is independent of the length scale parameter ϵ, Bauman-Phillips-Wang also demonstrated a uniform convergence up to the boundary. Some results on the region surrounding a spherical colloidal particle are also obtained by Alama-Bronsard-Lamy in [3]. Still under non-degenerate weak anchoring conditions, the Saturn ring defect was verified in the Landau-de Gennes theory.
In this article, we adopt the degenerate weak planar anchoring introduced in Fournier-Galatola [20]. The surface energy density is quartic and favors the planar anchoring. Consequently, the limiting minimizers must possess discontinuity on the boundary. A theoretical investigation in this direction was carried out by Alama-Bronsard-Golovaty in [2]. They studied the Ginzburg-Landau energy in a 2D thin-film domain. To promote oblique alignments, they introduced a surface energy density up to the fourth order. The authors identified a critical rate of convergence for the anchori
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