Author(s):
Costa, Fernando Pestana da 
; Grinfeld, Michael ; Mottram, Nigel J. ; Pinto, João Teixeira
Date: 2009
Persistent ID: http://hdl.handle.net/10400.2/1468
Origin: Repositório Aberto da Universidade Aberta
Subject(s): Freedericksz transition; Burgers’ equation; Convexity; Non-linear boundary value problems; Uniqueness of solutions
Description
In this paper we consider a boundary value problem for a quasilinear
pendulum equation with non-linear boundary conditions
that arises in a classical liquid crystals setup, the Freedericksz transition,
which is the simplest opto-electronic switch, the result of
competition between reorienting effects of an applied electric field
and the anchoring to the bounding surfaces. A change of variables
transforms the problem into the equation xττ = −f (x) for
τ ∈ (−T , T ), with boundary conditions xτ = ±βT
f (x) at τ = ∓T ,
for a convex non-linearity f . By analysing an associated inviscid
Burgers’ equation, we prove uniqueness of monotone solutions in
the original non-linear boundary value problem.
This result has been for many years conjectured in the liquid crystals
literature, e.g. in [E.G. Virga, Variational Theories for Liquid
Crystals, Appl. Math. Math. Comput., vol. 8, Chapman & Hall, London,
1994] and in [I.W. Stewart, The Static and Dynamic Continuum
Theory of Liquid Crystals: A Mathematical Introduction, Taylor &
Francis, London, 2003].
