Uniqueness in the Freedericksz transition with weak anchoring

Costa, Fernando Pestana da; Grinfeld, Michael; Mottram, Nigel J.; Pinto, João Teixeira

http://hdl.handle.net/10400.2/1468

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Authors

Costa, Fernando Pestana da

Grinfeld, Michael

Mottram, Nigel J.

Pinto, João Teixeira

Abstract(s)

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].

Keywords

Freedericksz transition Burgers’ equation Convexity Non-linear boundary value problems Uniqueness of solutions

URI

http://hdl.handle.net/10400.2/1468

Citation

Costa, Fernando Pestana da [et al.] - Uniqueness in the Freedericksz transition with weak anchoring. "Journal of Differential Equations" [Em linha]. ISSN 0022-0396. Nº 246 (2009), p. 2590-2600