Reduced Landau-de Gennes functional and surface smectic state of liquid crystals

In [X.B. Pan, Landau-de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1-2) (2003) 343-382], based on the de Gennes analogy between liquid crystals and superconductivity [P.G. de Gennes, An analogy between superconductors and smectics A, Solid State Commun. 10 (1972) 753-756], the second author introduced the critical wave number Qc₃ (which is an analog of the upper critical field Hc₃ for superconductors) and predicted the existence of a surface smectic state, which was supposed to be an analogy of the surface superconducting state. In a surface smectic state, the bulk liquid crystal is in the nematic state, and a thin layer of smectic appears in a helical strip on the surface of the sample. In this paper we study an approximate form of the Landau-de Gennes model of liquid crystals, and examine the behavior of minimizers, in particular the boundary layer behavior. Our work shows the importance of the joint chirality constant qτ, which is the product of wave number q and chirality τ and also appears in the work of [P. Bauman, M. Calderer, C. Liu, D. Phillips, The phase transition between chiral nematic and smectic A^∗ liquid crystals, Arch. Rational Mech. Anal. 165 (2002) 161-186] and [X.B. Pan, Landau-de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1-2) (2003) 343-382]. The joint chirality constant of a liquid crystal is useful to predict whether the liquid crystal is of type I or type II, and it is also useful to examine whether the liquid crystal is in a surface smectic state. The results in this paper suggest that a liquid crystal with large Ginzburg-Landau parameter κ and large joint chirality constant qτ exhibits type II behavior, and it will be in the surface smectic state if qτ∼bκ² for some β₀<b<1, where β₀ is the lowest eigenvalue of the Schrödinger operator with a unit magnetic field in the half space, and 0<β₀<1. We also show that a liquid crystal with small qτ exhibits type I behavior.     

Deformation of striped patterns by inhomogeneities

We study the effects of adding a local perturbation in a pattern-forming system, taking as an example the Ginzburg–Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a periodic pattern, one finds an unbounded linear operator that is not Fredholm due to continuous spectrum in typical translation invariant or weighted spaces. We show that Kondratiev spaces, which encode algebraic localization that increases with each derivative, provide an effective means to circumvent this difficulty. We establish Fredholm properties in such spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. We find a logarithmic phase correction, which vanishes for a particular spatial shift only, which we interpret as a phase-selection mechanism through the inhomogeneity. Copyright © 2013 John Wiley & Sons, Ltd.     

Diffuse interface methods for multiclass segmentation of high-dimensional data

We present two graph-based algorithms for multiclass segmentation of high-dimensional data, motivated by the binary diffuse interface model. One algorithm generalizes Ginzburg–Landau (GL) functional minimization on graphs to the Gibbs simplex. The other algorithm uses a reduction of GL minimization, based on the Merriman–Bence–Osher scheme for motion by mean curvature. These yield accurate and efficient algorithms for semi-supervised learning. Our algorithms outperform existing methods, including supervised learning approaches, on the benchmark datasets that we used. We refer to Garcia-Cardona (2014) for a more detailed illustration of the methods, as well as different experimental examples.     

Higher dimensional Ginzburg–Landau equations under weak anchoring boundary conditions

For $n\ge 3$ and $0<?\le 1$, let $\mathrm{\Omega }?{\mathbb{R}}^n$ be a bounded, simply connected, smooth domain, and $u_{?}:\mathrm{\Omega }?{\mathbb{R}}^n\to {\mathbb{R}}^2$ solve the Ginzburg–Landau equation under the weak anchoring boundary condition:

$\{\begin{array}{cc}?\mathrm{\Delta }{u}_{?}=\frac{1}{{?}^2}(1?|u_{?}{|}^2)u_{?}\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ \frac{?u_{?}}{?\nu }+{\lambda }_{?}(u_{?}?g_{?})=0\hfill & \mathrm{on}?\mathrm{\Omega },\hfill \end{array}$

where the anchoring strength parameter ${\lambda }_{?}=K{?}^{?\alpha }$ for some $K>0$ and $\alpha \in [0,1)$, and $g_{?}\in C^2(?\mathrm{\Omega },{\mathbb{S}}^1)$. Motivated by the connection with the Landau–De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the asymptotic behavior of $u_{?}$ as ? goes to zero under the condition that the total modified Ginzburg–Landau energy satisfies $F_{?}(u_{?},\mathrm{\Omega })\le M|\log ??|$ for some $M>0$.     

On weak solution to the steady compressible flow of nematic liquid crystals

In this paper, we study the three‐dimensional‐simplified Ericksen‐Leslie system for the steady compressible flow of nematic liquid crystals in a bounded domain. It is proved that the existence of a weak solution for the adiabatic exponent γ > 1 provided the initial direction field in the upper hemisphere.     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence of strong solutions to a time‐dependent 3D Ginzburg‐Landau model for superconductivity with partial viscous terms

We study an initial boundary value problem for a time‐dependent 3D Ginzburg‐Landau model of superconductivity with partial viscous terms. We prove the global existence of strong solutions.     

Minimum energy with infinite horizon: From stationary to non-stationary states

We study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state $\bar{x}=0$ (at time $t=-\infty$) into an arbitrary non-stationary state $x$ (at time $t=0$). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state $x$ into the equilibrium state $\bar{x}=0$). Consequently, the Algebraic Riccati Equation (ARE) associated with this problem is non-standard since the sign of the linear part is opposite to the usual one and since its solution is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper (Acquistapace and Gozzi, 2017). Here, similarly to such paper, we prove that the linear selfadjoint operator associated with the value function is a solution of the above mentioned ARE. Moreover, differently to Acquistapace and Gozzi (2017), we prove that such solution is the maximal one. The first main result (Theorem 5.8) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in Acquistapace and Gozzi (2017)). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem 6.5) and we apply this to the Landau–Ginzburg model.     

Studies of layering and chirality of smectic A* liquid crystals

We discuss mathematical and physical aspects of the phase transition from nematic to smectic A liquid crystals. The first approach deals with analyzing a model obtained from the Maier-Saupe theory of nematic by taking into account that elongated liquid crystal molecules present distinguishable ends. Moreover, we represent long range microscopic interactions by means of nonlocal free energy functionals. The smectic configurations emerge as solutions of the extended nematic theory, through a modulation process. The second part of the article deals with energy minimization of the de Gennes free energy for smectic A* liquid crystals, and with the study of uniform twist grain boundary (TGB) structures. The goal is to mathematically justify parameter regions of the phase diagram of the transition between nematic and smectic A liquid crystals. Both approaches complement each other from the point of view that, while the first one deals with mechanisms causing layer arrangements, the second approach focuses on how chirality and layer effects interact, in a system with preassumed periodicity. The A* notation refers to chiral liquid crystals.     

Freedericksz transitions in circular toroidal layers of smectic C liquid crystals

The aim of this paper is to consider theoretically a Freedericksztransition for concentric toroidal layers of smectic C liquidcrystal arising from a simple geometric setup, thereby extendingthe results of Atkin & Stewart [Q. Jl Mech. Appl. Math.,47, 1994] who considered spherical layers of smectic C in theusual cone and plate geometry. Application of smectic continuumtheory leads, after suitable approximations are made, to a lineargoverning equilibrium equation which is satisfied by both thetrivial solution and a variable solution involving Bessel functions.We are able to determine the critical magnitude _cH of the magneticfield H at which this variable solution exists, and a standardenergy comparison reveals that the variable solution is expectedto be more energetically favourable than the zero solution providedH > _cH. Numerical examples of critical thresholds are given,which are comparable to those in the literature for nematics.The paper ends with a discussion section and some indicationof possible future work.     

Ginzburg–Landau equations on Riemann surfaces of higher genus

We study the Ginzburg–Landau equations on Riemann surfaces of arbitrary genus. In particular, we

• –construct explicitly the (local moduli space of gauge-equivalent) solutions in the neighborhood of the constant curvature ones;
• –classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel–Jacobi map, and symmetric products of the surface;
• –determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions.

     

Existence and stability of coexistence states in a competition unstirred chemostat

In this paper, we consider the two similar competing species in a competition unstirred chemostat model with diffusion. The two competing species are assumed to be identical except for their maximal growth rates. In particular, we study the existence and stability of the coexistence states, and the semi-trivial equilibria or the unique coexistence state is the global attractor can be established under some suitable conditions. Our mathematical approach is based on Lyapunov–Schmidt reduction, the implicit function theory and spectral theory.     

Existence and asymptotic behavior of ground states for quasilinear singular equations involving Hardy-Sobolev exponents

We study the existence and decaying rate of solutions for the quasilinear problem

     

Existence and concentration of ground states of coupled nonlinear Schrödinger equations

In this paper we consider the existence and concentration of ground states of coupled nonlinear Schrödinger equations with trap potentials. When the interaction between two states is repulsive, we prove the existence of ground states. Then concentration phenomenon of these ground states is studied as the small perturbed parameter (Planck constant) approaches zero. Roughly speaking, we prove that components of the ground states concentrate at the unique global minimum points of their potentials. Moreover, we prove the existence of ground states when the interaction is attractive.