Uniqueness in the Freedericksz transition with weak anchoring

• Autor(a) principal: Costa, Fernando Pestana da
• Data de Publicação: 2009
• Outros Autores: Grinfeld, Michael, Mottram, Nigel J., Pinto, João Teixeira
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.2/1468
• topic: Freedericksz transition; Burgers’ equation; Convexity; Non-linear boundary value problems; Uniqueness of solutions

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].