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

Existence of solutions to boundary value problems for smectic liquid crystals

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.
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
References:
[1]

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

[3]

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

[4]

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

[5]

Eur. Phys. J. B, 6 (1998), 347-353. Google Scholar

[6]

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

[7]

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

[8]

Phys. Rev. A, 14 (1976), p. 1202. doi: 10.1103/PhysRevA.14.1202.  Google Scholar

[9]

Science, 301 (2003), 1204-1211. doi: 10.1126/science.1084956.  Google Scholar

[10]

Solid State Commun., 10 (1972), 753-756. Google Scholar

[11]

Clarendon Press, Oxford, 1993. Google Scholar

[12]

J. Phys.: Condensed Matter, 20 (2008), 1-9. Google Scholar

[13]

John Wiley $&$ Sons, Inc., 1984.  Google Scholar

[14]

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

[15]

Phys. Rev. A, 45 (1992), 5783-5788. doi: 10.1103/PhysRevA.45.5783.  Google Scholar

[16]

Springer, 2007. Google Scholar

[17]

Comm. Math Phys., 269 (2007), 369-399. doi: 10.1007/s00220-006-0132-z.  Google Scholar

[18]

Phys. Rev. E, 50 (1994), p. 2940. doi: 10.1103/PhysRevE.50.2940.  Google Scholar

[19]

Wiley-VCH, 1999. Google Scholar

[20]

Phys. Rev. E, 41 (1990), p. 4392. doi: 10.1103/PhysRevA.41.4392.  Google Scholar

[21]

Phys. Rev. E, 57 (1994), 574-581. doi: 10.1103/PhysRevE.57.574.  Google Scholar

[22]

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

[23]

J. Phys. II France, 5 (1995), 1223-1240. Google Scholar

[24]

SIAM J. Appl. Math., 66 (2006), 2107-2126. doi: 10.1137/050641120.  Google Scholar

[25]

Phys. Rev. A, 45 (1992), p. 953. doi: 10.1103/PhysRevA.45.953.  Google Scholar

[26]

Phys. Rev. E, 58 (1998), p. 7455. Google Scholar

[27]

Phys. Rev. E, 63 (2001), p. 031705. doi: 10.1103/PhysRevE.63.031705.  Google Scholar

[28]

Phys. Rev. E, 68 (2003), p. 061705. Google Scholar

[29]

Phys. Rev. Lett., 98 (2007), p. 247802. Google Scholar

[30]

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

[31]

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

show all references

References:
[1]

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

[3]

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

[4]

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

[5]

Eur. Phys. J. B, 6 (1998), 347-353. Google Scholar

[6]

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

[7]

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

[8]

Phys. Rev. A, 14 (1976), p. 1202. doi: 10.1103/PhysRevA.14.1202.  Google Scholar

[9]

Science, 301 (2003), 1204-1211. doi: 10.1126/science.1084956.  Google Scholar

[10]

Solid State Commun., 10 (1972), 753-756. Google Scholar

[11]

Clarendon Press, Oxford, 1993. Google Scholar

[12]

J. Phys.: Condensed Matter, 20 (2008), 1-9. Google Scholar

[13]

John Wiley $&$ Sons, Inc., 1984.  Google Scholar

[14]

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

[15]

Phys. Rev. A, 45 (1992), 5783-5788. doi: 10.1103/PhysRevA.45.5783.  Google Scholar

[16]

Springer, 2007. Google Scholar

[17]

Comm. Math Phys., 269 (2007), 369-399. doi: 10.1007/s00220-006-0132-z.  Google Scholar

[18]

Phys. Rev. E, 50 (1994), p. 2940. doi: 10.1103/PhysRevE.50.2940.  Google Scholar

[19]

Wiley-VCH, 1999. Google Scholar

[20]

Phys. Rev. E, 41 (1990), p. 4392. doi: 10.1103/PhysRevA.41.4392.  Google Scholar

[21]

Phys. Rev. E, 57 (1994), 574-581. doi: 10.1103/PhysRevE.57.574.  Google Scholar

[22]

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

[23]

J. Phys. II France, 5 (1995), 1223-1240. Google Scholar

[24]

SIAM J. Appl. Math., 66 (2006), 2107-2126. doi: 10.1137/050641120.  Google Scholar

[25]

Phys. Rev. A, 45 (1992), p. 953. doi: 10.1103/PhysRevA.45.953.  Google Scholar

[26]

Phys. Rev. E, 58 (1998), p. 7455. Google Scholar

[27]

Phys. Rev. E, 63 (2001), p. 031705. doi: 10.1103/PhysRevE.63.031705.  Google Scholar

[28]

Phys. Rev. E, 68 (2003), p. 061705. Google Scholar

[29]

Phys. Rev. Lett., 98 (2007), p. 247802. Google Scholar

[30]

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

[31]

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

[1]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003

[2]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398

[3]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019

[4]

Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021094

[5]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[6]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021092

[7]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[8]

Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021049

[9]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[10]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637

[11]

Guanming Gai, Yuanyuan Nie, Chunpeng Wang. A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021070

[12]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453

[13]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083

[14]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[15]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[16]

Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021043

[17]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019

[18]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[19]

Hongjie Dong, Xinghong Pan. On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021049

[20]

Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]