### Abstract

Language | English |
---|---|

Pages | 2590-2600 |

Number of pages | 11 |

Journal | Journal of Differential Equations |

Volume | 246 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Apr 2009 |

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### Keywords

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

### Cite this

*Journal of Differential Equations*,

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

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*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).

Research output: Contribution to journal › Article

TY - JOUR

T1 - Uniqueness in the Freedericksz transition with weak anchoring

AU - Da Costa, F.P.

AU - Grinfeld, M.

AU - Mottram, N.J.

AU - Pinto, J.T.

AU - FCT Portugal (partly supported) (Funder)

PY - 2009/4/1

Y1 - 2009/4/1

N2 - 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.

AB - 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.

KW - Freedericksz transition

KW - Burgers' equation

KW - convexity

KW - nonlinear boundar

KW - uniqueness of solutions

KW - value problems

U2 - 10.1016/j.jde.2009.01.033

DO - 10.1016/j.jde.2009.01.033

M3 - Article

VL - 246

SP - 2590

EP - 2600

JO - Journal of Differential Equations

T2 - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 7

ER -