Symmetry and Multiplicity of Solutions in a Two-Dimensional Landau–de Gennes Model for Liquid Crystals

We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R. We prove that under a symmetric boundary condition carrying a topological defect of degree k 2 for some given even non-zero integer k, there are exactly two minimizers for all large enough R. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points

Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals

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. We investigate in greater detail the regime of vanishing elastic constant $L \to 0$, where we obtain three explicit point defect profiles, including the global minimiser.Comment: 15 pages, 16 figure

Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals

We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of Γ-development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing Q-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy

Orientability and energy minimization in liquid crystal models

Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a unit vector field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class $W^{1,2}$ the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains we completely characterise the instances in which the predictions of the constrained Landau-de Gennes theory differ from those of the Oseen-Frank theory

Induction of differentiation inhibits the tumorigenic potential of glioblastoma cancer stem cells

The outcome of the patients with newly diagnosed glioblastoma remains dismal, despite the use of surgery, radiotherapy and adjuvant temozolomide and while new agents like anti-angiogenic agents seem to offer some promise, a new approach is needed. Recent studies suggest that cancer stem cells (CSCs) may play an important role in malignant gliomas invasion and proliferation. Therefore, CSCs became new therapeutical targets, and one of the main experimental therapies which could be used against CSCs is the differentiation therapy.The purpose of this study was to characterize the CSCs isolated from glioblastoma samples, to assess in vivo the tumorigenic potential of these cells and to induct the differentiation of the CSCs. The changes in invasive markers (matrixmetalloproteases-MMPs, cadherins and cathenins) expression were assessed. CSCs exposed to differentiation inductor factors have been inoculated in nude mice and their tumorigenic potential has been evaluated. The stemness biological feature was correlated with increased of MMPs, cadherins, catenin expression and with tumour contra-lateral invasion. The expression of MMPs, cadherins and cadherins decreased after exposure of the CSCs cultures to the differentiation inductor factors. In vivo experiments demonstrated the inhibition of tumorigenic potential of differentiated CSCs cultures.In conclusion, differentiated CSCs showed a decreased expression of invasive markers in vitro and lost their tumorigenic potential in vivo

Common genetic variants in the CLDN2 and PRSS1-PRSS2 loci alter risk for alcohol-related and sporadic pancreatitis

Pancreatitis is a complex, progressively destructive inflammatory disorder. Alcohol was long thought to be the primary causative agent, but genetic contributions have been of interest since the discovery that rare PRSS1, CFTR, and SPINK1 variants were associated with pancreatitis risk. We now report two significant genome-wide associations identified and replicated at PRSS1-PRSS2 (1×10-12) and x-linked CLDN2 (p < 1×10-21) through a two-stage genome-wide study (Stage 1, 676 cases and 4507 controls; Stage 2, 910 cases and 4170 controls). The PRSS1 variant affects susceptibility by altering expression of the primary trypsinogen gene. The CLDN2 risk allele is associated with atypical localization of claudin-2 in pancreatic acinar cells. The homozygous (or hemizygous male) CLDN2 genotype confers the greatest risk, and its alleles interact with alcohol consumption to amplify risk. These results could partially explain the high frequency of alcohol-related pancreatitis in men – male hemizygous frequency is 0.26, female homozygote is 0.07

On a sharp Poincare-type inequality on the 2-sphere and its application in micromagnetics

The main aim of this note is to prove a sharp Poincare-type inequality for vector-valued functions on S2 that naturally emerges in the context of micromagnetics of spherical thin films

Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals

We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R. We prove that under a symmetric boundary condition carrying a topological defect of degree 2 for some given even non-zero integer k, there are exactly two minimizers for all large enough R. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points

On the uniqueness of minimisers of Ginzburg-Landau functionals: Sur l'unicité des minimiseurs de la fonctionnelle de Ginzburg-Landau

We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for Rn-valued maps under a suitable convexity assumption on the potential and for H1/2 ∩ L1 boundary data that is non-negative in a fixed direction e ∈ Sn−1. Furthermore, we show that, when minimisers are not unique, the set of minimisers is generated from any of its elements using appropriate orthogonal transformations of Rn. We also prove corresponding results for harmonic maps