### 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 language | English |
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Pages (from-to) | 2590-2600 |

Number of pages | 11 |

Journal | Journal of Differential Equations |

Volume | 246 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Apr 2009 |

### Keywords

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

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## Cite this

Da Costa, F. P., Grinfeld, M., Mottram, N. J., Pinto, J. T., & FCT Portugal (partly supported) (Funder) (2009). Uniqueness in the Freedericksz transition with weak anchoring.

*Journal of Differential Equations*,*246*(7), 2590-2600. https://doi.org/10.1016/j.jde.2009.01.033