The Saturn Ring Effect in Nematic Liquid Crystals with External Field: Effective Energy and Hysteresis

Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field

“…Coupled with this optimization problem, we show using a Modica-Mortola type argument that the remaining part of the energy concentrates on hypersurfaces which end either on the boundary of the domain or on the described line. This article is the continuation of the work started in [4], our main theorem (Theorem 3.1) is a generalisation of Theorem 3.1 in [4] (see Remark 3.3). In particular, our new theorem holds for an arbitrary manifold M of class C 1,1 instead of a sphere and we remove the hypothesis of rotational equivariance of Q.…”

“…It can be checked that f given in (2) satisfies these assumptions [4,12,14,46]. The exponent 4 in the last assumption is not crucial but only needs to outweigh the growth of g.…”

Pattern Formation in Landau-de Gennes Theory

“…Let w a denote either w + a,b or w − a,c . Then we have (1). There exist a 0 > 0 suitably large and δ 0 ∈ 0, 1/4 so that all zeros of w a are contained in l + δ0 l − δ0 , provided that a > a 0 .…”

“…Note that solutions to the LdG equation with ring-like disclinations have also been considered in [10,11] when the order parameter Q satisfies the Lyuksyutov constraint. Interested readers may also refer to [4,5,19,7,20,3,13,21,8,9,18,1] for various recent studies on solutions to the LdG equation with disclinations.…”

“…Due to Lemma 3.10 and the limit a n λ n r n 2 ∞ , we have the following modification of Lemma 3.3: Lemma 3.11. There exist an increasing positive sequence σ k tending to 1 as k ∞, a sequence of positive numbers b k and a subsequence of U n , still denoted by U n , so that for any k, (1). The mappings U n , U sc n and U sc ∞ satisfy the uniform boundedness given below:…”