54 research outputs found
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 . 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 -harmonic maps from three-dimensional domains to
the real projective plane, for . 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 -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 , 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 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
Line defects in the vanishing elastic constant limit of a three-dimensional Landau-de Gennes model
62 pages, 4 figures.We consider the Landau-de Gennes variational model for nematic liquid crystals, in three-dimensional domains. We are interested in the asymptotic behaviour of minimizers as the elastic constant tends to zero. Assuming that the energy of minimizers is bounded by the logarithm of the elastic constant, there exists a relatively closed, 1-rectiable set S line of nite length, such that minimizers converge to a locally harmonic map away from S line. We provide sucient conditions for the logarithmic energy bound to be satised. Finally, we show by an example that the limit map may have both point and line singularities
Polydispersity and surface energy strength in nematic colloids
We consider a Landau-de Gennes model for a polydisperse, inhomogeneous
suspension of colloidal inclusions in a nematic host, in the dilute regime. We
study the homogenised limit and compute the effective free energy of the
composite material. By suitably choosing the shape of the inclusions and
imposing a quadratic, Rapini-Papoular type surface anchoring energy density, we
obtain an effective free energy functional with an additional linear term,
which may be interpreted as an "effective field" induced by the inclusions.
Moreover, we compute the effective free energy in a regime of "very strong
anchoring", that is, when the surface energy effects dominate over the volume
free energy.Comment: 24 pages, 1 figur
Dynamics of Ginzburg-Landau vortices for vector fields on surfaces
In this paper we consider the gradient flow of the following Ginzburg-Landau
type energy
This energy is defined on tangent vector fields on a -dimensional closed and
oriented Riemannian manifold (here stands for the covariant derivative)
and depends on a small parameter . If the energy satisfies
proper bounds, when the second term forces the vector fields
to have unit length. However, due to the incompatibility for vector fields on
between the Sobolev regularity and the unit norm constraint, critical
points of tend to generate a finite number of singular points
(called vortices) having non-zero index (when the Euler characteristic is
non-zero). These types of problems have been extensively analyzed in a recent
paper by R. Ignat and R. Jerrard. As in Euclidean case, the position of the
vortices is ruled by the so-called renormalized energy.
In this paper we are interested in the dynamics of vortices. We rigorously
prove that the vortices move according to the gradient flow of the renormalized
energy, which is the limit behavior when of the gradient
flow of the Ginzburg-Landau energy.Comment: 71 pages, 1 figur
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