Uniqueness in the Freedericksz transition with weak anchoring

F.P. Da Costa, M. Grinfeld, N.J. Mottram, J.T. Pinto, FCT Portugal (partly supported) (Funder)

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Abstract

In this paper we consider a boundary value problem for a quasilinear pendulum equation with nonlinear 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 x = −f(x) for ∈ (−T, T), with boundary conditions x = ± T f(x) at = ∓T, for a convex nonlinearity f. By analyzing an associated inviscid Burgers' equation, we prove uniqueness of monotone solutions in the original nonlinear 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,Chapman and Hall, London, 1994 and in I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor and Francis, London, 2003.
Original languageEnglish
Pages (from-to)2590-2600
Number of pages11
JournalJournal of Differential Equations
Volume246
Issue number7
DOIs
Publication statusPublished - 1 Apr 2009

Keywords

  • Freedericksz transition
  • Burgers' equation
  • convexity
  • nonlinear boundar
  • uniqueness of solutions
  • value problems

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