American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - B, 2009, 11 (1) : 153-176. doi: 10.3934/dcdsb.2009.11.153
[1]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623
[2]

David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311
[3]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[4]

Carlos J. García-Cervera, Sookyung Joo. Reorientation of smectic a liquid crystals by magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1983-2000. doi: 10.3934/dcdsb.2015.20.1983
[5]

Jinhae Park, Feng Chen, Jie Shen. Modeling and simulation of switchings in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1419-1440. doi: 10.3934/dcds.2010.26.1419
[6]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
[7]

Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 853-865. doi: 10.3934/cpaa.2008.7.853
[8]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
[9]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[10]

Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445
[11]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757
[12]

Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499
[13]

Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243
[14]

Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
[15]

Xian-Gao Liu, Jianzhong Min, Kui Wang, Xiaotao Zhang. Serrin's regularity results for the incompressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5579-5594. doi: 10.3934/dcds.2016045
[16]

Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1
[17]

Tiziana Giorgi, Feras Yousef. Analysis of a model for bent-core liquid crystals columnar phases. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2001-2026. doi: 10.3934/dcdsb.2015.20.2001
[18]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357
[19]

Shijin Ding, Junyu Lin, Changyou Wang, Huanyao Wen. Compressible hydrodynamic flow of liquid crystals in 1-D. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 539-563. doi: 10.3934/dcds.2012.32.539
[20]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences