Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model

We consider the Landau-de Gennes variational model for nematic liquid crystals, in three-dimensional domains. More precisely, we study the asymptotic behaviour of minimizers as the elastic constant tends to zero, under the assumption that minimizers are uniformly bounded and their energy blows up as the logarithm of the elastic constant. We show that there exists a closed set S of finite length, such that minimizers converge to a locally harmonic map away from S. Moreover, S restricted to the interior of the domain is a locally finite union of straight line segments. We provide sufficient conditions, depending on the domain and the boundary data, under which our main results apply. We also discuss some examples.Comment: 71 pages, 5 figure

Biaxiality in the asymptotic analysis of a 2-D Landau-de Gennes model for liquid crystals

We consider the Landau-de Gennes variational problem on a bound\-ed, two dimensional domain, subject to Dirichlet smooth boundary conditions. We prove that minimizers are maximally biaxial near the singularities, that is, their biaxiality parameter reaches the maximum value $1$. Moreover, we discuss the convergence of minimizers in the vanishing elastic constant limit. Our asymptotic analysis is performed in a general setting, which recovers the Landau-de Gennes problem as a specific case.Comment: 34 pages, 2 figures; typos corrected, minor changes in proofs. Results are unchange

Improved partial regularity for manifold-constrained minimisers of subquadratic energies

We consider minimising $p$-harmonic maps from three-dimensional domains to the real projective plane, for $1<p<2$. These maps arise as least-energy configurations in variational models for nematic liquid crystals. We show that the singular set of such a map decomposes into a $1$-dimensional set, which can be physically interpreted as a non-orientable line defect, and a locally finite set, i.e. a collection of point defects.Comment: New version: typos and inaccuracies fixe

Convergence properties for a generalization of the Caginalp phase field system

We are concerned with a phase field system consisting of two partial differential equations in terms of the variables thermal displacement, that is basically the time integration of temperature, and phase parameter. The system is a generalization of the well-known Caginalp model for phase transitions, when including a diffusive term for the thermal displacement in the balance equation and when dealing with an arbitrary maximal monotone graph, along with a smooth anti-monotone function, in the phase equation. A Cauchy-Neumann problem has been studied for such a system in arXiv:1107.3950v2 [math.AP], by proving well-posedness and regularity results, as well as convergence of the problem as the coefficient of the diffusive term for the thermal displacement tends to zero. The aim of this contribution is rather to investigate the asymptotic behaviour of the problem as the coefficient in front of the Laplacian of the temperature goes to 0: this analysis is motivated by the types III and II cases in the thermomechanical theory of Green and Naghdi. Under minimal assumptions on the data of the problems, we show a convergence result. Then, with the help of uniform regularity estimates, we discuss the rate of convergence for the difference of the solutions in suitable norms.Comment: Key words: phase field model, initial-boundary value problem, regularity of solutions, convergence, error estimate

Order Reconstruction for Nematics on Squares and Regular Polygons: A Landau-de Gennes Study

We construct an order reconstruction (OR)-type Landau-de Gennes critical point on a square domain of edge length $\lambda$, motivated by the well order reconstruction solution numerically reported by Kralj and Majumdar. The OR critical point is distinguished by an uniaxial cross with negative scalar order parameter along the square diagonals. The OR critical point is defined in terms of a saddle-type critical point of an associated scalar variational problem. The OR-type critical point is globally stable for small $\lambda$ and undergoes a supercritical pitchfork bifurcation in the associated scalar variational setting. We consider generalizations of the OR-type critical point to a regular hexagon, accompanied by numerical estimates of stability criteria of such critical points on both a square and a hexagon in terms of material-dependent constants.Comment: 29 pages, 12 figure

Changes in intracellular calcium and glutathione in astrocytes as the primary mechanism of amyloid neurotoxicity

Although the accumulation of the neurotoxic peptide {beta} amyloid ({beta}A) in the CNS is a hallmark of Alzheimer's disease, the mechanism of {beta}A neurotoxicity remains controversial. In cultures of mixed neurons and astrocytes, we found that both the full-length peptide {beta}A (1–42) and the neurotoxic fragment (25–35) caused sporadic cytoplasmic calcium [intracellular calcium ([Ca2+]c)] signals in astrocytes that continued for hours, whereas adjacent neurons were completely unaffected. Nevertheless, after 24 hr, although astrocyte cell death was marginally increased, ~50% of the neurons had died. The [Ca2+]c signal was entirely dependent on Ca2+ influx and was blocked by zinc and by clioquinol, a heavy-metal chelator that is neuroprotective in models of Alzheimer's disease. Neuronal death was associated with Ca2+-dependent glutathione depletion in both astrocytes and neurons. Thus, astrocytes appear to be the primary target of {beta}A, whereas the neurotoxicity reflects the neuronal dependence on astrocytes for antioxidant support

β-Amyloid peptides induce mitochondrial dysfunction and oxidative stress in astrocytes and death of neurons through activation of NADPH oxidase

β-Amyloid (βA) peptide is strongly implicated in the neurodegeneration underlying Alzheimer's disease, but the mechanisms of neurotoxicity remain controversial. This study establishes a central role for oxidative stress by the activation of NADPH oxidase in astrocytes as the cause of βA-induced neuronal death. βA causes a loss of mitochondrial potential in astrocytes but not in neurons. The mitochondrial response consists of Ca2+-dependent transient depolarizations superimposed on a slow collapse of potential. The slow response is both prevented by antioxidants and, remarkably, reversed by provision of glutamate and other mitochondrial substrates to complexes I and II. These findings suggest that the depolarization reflects oxidative damage to metabolic pathways upstream of mitochondrial respiration. Inhibition of NADPH oxidase by diphenylene iodonium or 4-hydroxy-3-methoxy-acetophenone blocks βA-induced reactive oxygen species generation, prevents the mitochondrial depolarization, prevents βA-induced glutathione depletion in both neurons and astrocytes, and protects neurons from cell death, placing the astrocyte NADPH oxidase as a primary target of βA-induced neurodegeneration