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)

Research output: Contribution to journalArticle

6 Citations (Scopus)

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.
LanguageEnglish
Pages2590-2600
Number of pages11
JournalJournal of Differential Equations
Volume246
Issue number7
DOIs
Publication statusPublished - 1 Apr 2009

Fingerprint

Liquid Crystal
Uniqueness
Change of Variables
Nonlinear Boundary Value Problems
Nonlinear Boundary Conditions
Optoelectronics
Pendulum
Burgers Equation
Electric Field
Switch
Monotone
Continuum
Boundary Value Problem
Nonlinearity
Transform
Boundary conditions

Keywords

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

Cite this

Da Costa, F.P. ; Grinfeld, M. ; Mottram, N.J. ; Pinto, J.T. ; FCT Portugal (partly supported) (Funder). / Uniqueness in the Freedericksz transition with weak anchoring. In: Journal of Differential Equations. 2009 ; Vol. 246, No. 7. pp. 2590-2600.
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Da Costa, FP, Grinfeld, M, Mottram, NJ, Pinto, JT & FCT Portugal (partly supported) (Funder) 2009, 'Uniqueness in the Freedericksz transition with weak anchoring' Journal of Differential Equations, vol. 246, no. 7, pp. 2590-2600. https://doi.org/10.1016/j.jde.2009.01.033

Uniqueness in the Freedericksz transition with weak anchoring. / Da Costa, F.P.; Grinfeld, M.; Mottram, N.J.; Pinto, J.T.; FCT Portugal (partly supported) (Funder).

In: Journal of Differential Equations, Vol. 246, No. 7, 01.04.2009, p. 2590-2600.

Research output: Contribution to journalArticle

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AU - Da Costa, F.P.

AU - Grinfeld, M.

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