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April  2015, 8(2): 243-257. doi: 10.3934/dcdss.2015.8.243

Existence of solutions to boundary value problems for smectic liquid crystals

Patricia Bauman 1, , Daniel Phillips 1, and  Jinhae Park 2,

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47906, United States
2. 

Department of Mathematics, Chungnam National University, 99 Daehak-ro, Gung-Dong Yuseong-gu, Daejeon 305-764

Received  April 2013 Revised  October 2013 Published  July 2014

We prove lower semicontinuity and lower bounds for a Chen-Lubensky energy describing nematic/smectic liquid crystals with physically realistic boundary conditions. The Chen-Lubensky energy captures stable phases of the liquid crystal material, ranging from purely nematic or smectic states to coexisting nematic/smectic states. By including appropriate additional terms, the model includes the effects of applied electric or magnetic fields, and/or electrical self-interactions in the case of polarized liquid crystals. As a consequence of our results, we establish existence of minimizers with weak or strong anchoring of the director field (describing molecular orientation) at the boundary, and Dirichlet or Neumann boundary conditions on the smectic order parameter for the liquid crystal material.
Keywords: ferroelectrics, Calculus of variations, boundary value problem., smectic liquid crystal, elliptic equations and systems.
Mathematics Subject Classification: Primary: 49J40, 35J50; Secondary: 35J57, 35J6.
Citation: 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
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show all references

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Phys. Rev. E, 75 (2007), 031701. doi: 10.1103/PhysRevE.75.031701.  Google Scholar

[2]

Arch. Rational Mech. Anal., 165 (2002), 161-186. doi: 10.1007/s00205-002-0223-8.  Google Scholar

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Molecular Crystals and Liquid Crystals Journal, 510 (2009), 1135-1145. Google Scholar

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Liquid Crystal Today, 9 (1999), 1-10. Google Scholar

[5]

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[6]

Phys. Rev. Lett., 70 (1993), p. 2742. doi: 10.1103/PhysRevLett.70.2742.  Google Scholar

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Phys. Rev. E, 53 (1996), 4933-4943. doi: 10.1103/PhysRevE.53.4933.  Google Scholar

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Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

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Springer, 2007. Google Scholar

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[22]

World-Scientific, Singapore, New Jersey, London, Hong Kong, 2000. Google Scholar

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[27]

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Phys. Rev. E, 62 (2000), p. 2317. Google Scholar

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Phys. Rev. E, 54 (1996), p. 3783. Google Scholar

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