Uniqueness in the Freedericksz transition with weak anchoring

Da Costa, F. P., Grinfeld, M., Mottram, N. J. and Pinto, J. T. (2009) Uniqueness in the Freedericksz transition with weak anchoring. Journal of Differential Equations, 246(7), pp. 2590-2600. (doi: 10.1016/j.jde.2009.01.033)

Abstract

In this paper we consider a boundary value problem for a quasi-linear 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=±βTf(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].

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Mottram, Professor Nigel
• Authors: Da Costa, F. P., Grinfeld, M., Mottram, N. J., and Pinto, J. T.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Differential Equations
• Publisher: Elsevier Inc.
• ISSN: 0022-0396
• ISSN (Online): 1090-2732
• Published Online: 11 February 2009