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January  2009, 11(1): 153-176. doi: 10.3934/dcdsb.2009.11.153

The Janossy effect and hybrid variational principles

David Kinderlehrer 1, and  Michał Kowalczyk 2,

1. 

Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890
2. 

Departamento de Ingenier´ıa Matem´atica and, Centro de Modelamiento Matem´atico (UMI 2807 CNRS), Universidad de Chile, FCFM, Santiago, Chile

Received  February 2008 Revised  May 2008 Published  November 2008

Light can change the orientation of a liquid crystal. This is the optical Freedericksz transition, discovered by Saupe. In the Janossy effect, the threshold intensity for the optical Freedericksz transition is dramatically reduced by the additon of a small amount of dye to the sample. This has been interpreted as an optically pumped orientational rachet mechanism, similar to the rachet mechanism in biological molecular motors. To interpret the evolution system proposed for this effect requires an innovative gradient flow. Here we introduce this gradient flow and illustrate how it also provides the boundary conditions, some unusual coupling conditions, between the liquid crystal and the dye. An existence theorem for the evolution problem follows as well. Furthermore, we consider the time independent problem and show its local asymptotic stability. Finally we progress toward showing that the proposed model correctly predicts the onset of the Janossy effect.
Keywords: liquid crystals, elliptic systems, Mass transport.
Mathematics Subject Classification: 35K45, 35K50, 35K55, 35K57, 92C37, 92C45.
Citation: David Kinderlehrer, Michał Kowalczyk. The Janossy effect and hybrid variational principles. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 153-176. doi: 10.3934/dcdsb.2009.11.153
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