scholarly journals Sphere-valued harmonic maps with surface energy and the K13 problem

2019 ◽  
Vol 12 (4) ◽  
pp. 363-392
Author(s):  
Stuart Day ◽  
Arghir Dani Zarnescu
Keyword(s):  
Liquid Crystals ◽  
Surface Energy ◽  
Boundary Conditions ◽  
Critical Point ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Nematic Liquid Crystals ◽  
Energy Functional ◽  
Dirichlet Energy ◽  
Surface Term

AbstractWe consider an energy functional motivated by the celebrated {K_{13}} problem in the Oseen–Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimisers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the global minimisers.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

scholarly journals STABILITY FOR NEMATIC LIQUID CRYSTALS WITH STRETCHING TERMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2937-2942 ◽  
Author(s):  
B. CLIMENT-EZQUERRA ◽  
F. GUILLÉN-GONZÁLEZ ◽  
M. A. RODRÍGUEZ-BELLIDO
Keyword(s):  
Liquid Crystals ◽  
Asymptotic Stability ◽  
Boundary Conditions ◽  
Present Article ◽  
Periodic Boundary ◽  
Nematic Liquid Crystals ◽  
Periodic Boundary Conditions ◽  
Stability Properties ◽  
Crystal Model ◽  
Different Shapes

We study a nematic crystal model that appeared in [Liu et al., 2007], modeling stretching effects depending on the different shapes of the microscopic molecules of the material, under periodic boundary conditions. The aim of the present article is two-fold: to extend the results given in [Sun & Liu, 2009], to a model with more complete stretching terms and to obtain some stability and asymptotic stability properties for this model.

Dissipation, surface energy and defects

1997 ◽  
Vol 8 (3) ◽  
pp. 301-310 ◽  
Author(s):  
M. CARME CALDERER
Keyword(s):  
Liquid Crystals ◽  
Surface Energy ◽  
Layered Structure ◽  
Poiseuille Flow ◽  
Nematic Liquid Crystals ◽  
Gradient Flows ◽  
Flow Problems ◽  
Constant Gradient

A model for Poiseuille flow of nematic liquid crystals is examined where a layered structure of defects parallel to the flow is present. Constant gradient flows are discussed, together with symmetric and chevron configurations for such flow problems.

scholarly journals Expulsion of disclinations in nematic liquid crystals

2003 ◽  
Vol 14 (1) ◽  
pp. 39-59 ◽  
Author(s):  
PAOLO BISCARI ◽  
TIMOTHY J. SLUCKIN
Keyword(s):  
Liquid Crystal ◽  
Liquid Crystals ◽  
Boundary Conditions ◽  
Half Space ◽  
Nematic Liquid Crystal ◽  
Nematic Liquid Crystals ◽  
Critical Value ◽  
Repulsive Force ◽  
Anchoring Strength ◽  
Strong Anchoring

We study the interactions between a nematic liquid crystal disclination and the surface of the half-space which bounds it. When strong anchoring conditions are applied on the boundary, the biaxial core of the disclination affects the repulsive force that tends to drive the disclination away from the surface. If we replace the strong boundary conditions with an anchoring potential, the surface-disclination interaction depends on the surface extrapolation length. In particular, the nematic may expel the disclination if the anchoring strength is below a critical value.

Sign in / Sign up

Export Citation Format

Share Document