1,281 research outputs found
Orientability and energy minimization in liquid crystal models
Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory
through a unit vector field . This theory has the apparent drawback that it
does not respect the head-to-tail symmetry in which should be equivalent to
-. This symmetry is preserved in the constrained Landau-de Gennes theory
that works with the tensor .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
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
Energies of S^2-valued harmonic maps on polyhedra with tangent boundary conditions
A unit-vector field n:P \to S^2 on a convex polyhedron P \subset R^3
satisfies tangent boundary conditions if, on each face of P, n takes values
tangent to that face. Tangent unit-vector fields are necessarily discontinuous
at the vertices of P. We consider fields which are continuous elsewhere. We
derive a lower bound E^-_P(h) for the infimum Dirichlet energy E^inf_P(h) for
such tangent unit-vector fields of arbitrary homotopy type h. E^-_P(h) is
expressed as a weighted sum of minimal connections, one for each sector of a
natural partition of S^2 induced by P. For P a rectangular prism, we derive an
upper bound for E^inf_P(h) whose ratio to the lower bound may be bounded
independently of h. The problem is motivated by models of nematic liquid
crystals in polyhedral geometries. Our results improve and extend several
previous results.Comment: 42 pages, 2 figure
On mathematical models for Bose-Einstein condensates in optical lattices (expanded version)
Our aim is to analyze the various energy functionals appearing in the physics
literature and describing the behavior of a Bose-Einstein condensate in an
optical lattice. We want to justify the use of some reduced models. For that
purpose, we will use the semi-classical analysis developed for linear problems
related to the Schr\"odinger operator with periodic potential or multiple wells
potentials. We justify, in some asymptotic regimes, the reduction to low
dimensional problems and analyze the reduced problems
Eigenelements of a General Aggregation-Fragmentation Model
We consider a linear integro-differential equation which arises to describe
both aggregation-fragmentation processes and cell division. We prove the
existence of a solution (\lb,\U,\phi) to the related eigenproblem. Such
eigenelements are useful to study the long time asymptotic behaviour of
solutions as well as the steady states when the equation is coupled with an
ODE. Our study concerns a non-constant transport term that can vanish at
since it seems to be relevant to describe some biological processes like
proteins aggregation. Non lower-bounded transport terms bring difficulties to
find estimates. All the work of this paper is to solve this problem
using weighted-norms
Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups
26 page
Time--Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System
Several singular limits are investigated in the context of a
system arising for instance in the modeling of chromatographic processes. In
particular, we focus on the case where the relaxation term and a
projection operator are concentrated on a discrete lattice by means of Dirac
measures. This formulation allows to study more easily some time-splitting
numerical schemes
Linear -positive sets and their polar subspaces
In this paper, we define a Banach SNL space to be a Banach space with a
certain kind of linear map from it into its dual, and we develop the theory of
linear -positive subsets of Banach SNL spaces with Banach SNL dual spaces.
We use this theory to give simplified proofs of some recent results of
Bauschke, Borwein, Wang and Yao, and also of the classical Brezis-Browder
theorem.Comment: 11 pages. Notational changes since version
Classification of unit-vector fields in convex polyhedra with tangent boundary conditions
A unit-vector field n on a convex three-dimensional polyhedron P is tangent
if, on the faces of P, n is tangent to the faces. A homotopy classification of
tangent unit-vector fields continuous away from the vertices of P is given. The
classification is determined by certain invariants, namely edge orientations
(values of n on the edges of P), kink numbers (relative winding numbers of n
between edges on the faces of P), and wrapping numbers (relative degrees of n
on surfaces separating the vertices of P), which are subject to certain sum
rules. Another invariant, the trapped area, is expressed in terms of these. One
motivation for this study comes from liquid crystal physics; tangent
unit-vector fields describe the orientation of liquid crystals in certain
polyhedral cells.Comment: 21 pages, 2 figure
A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces
We consider the scalar semilinear heat equation ut−Δu=f(u), where f:[0,∞)→[0,∞) is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in Lq(Ω) for all non-negative initial data u0∈Lq(Ω), when Ω⊂Rd is a bounded domain with Dirichlet boundary conditions. For q∈(1,∞) this holds if and only if limsups→∞s−(1+2q/d)f(s
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