Refined approximation for minimizers of a Landau-de Gennes energy functional

We study minimizers of a Landau-de Gennes energy functional in the asymptotic regime of small dimensionless elastic constant L > 0. The results on the convergence to a minimizer of the limit Oseen-Frank functional in Majumdar and Zarnescu (Arch Ration Mech Anal 196:227–280, 2010) are revisited and improved, which in effect lead to a sharp rate of convergence. The equation for the first-order correction term is derived: it has a “normal component” given by an algebraic relation and a “tangential component” given by a linear system.

[1]  E. Virga,et al.  Universal fine structure of nematic hedgehogs , 2001 .

[2]  F. Lin,et al.  Evolution of harmonic maps with Dirichlet boundary conditions , 1993 .

[3]  Frédéric Hélein,et al.  Asymptotics for the minimization of a Ginzburg-Landau functional , 1993 .

[4]  P. Sheng,et al.  Introduction to Liquid Crystals , 1976 .

[5]  L. Nikolova,et al.  On ψ- interpolation spaces , 2009 .

[6]  Frédéric Hélein,et al.  Harmonic Maps, Conservation Laws, And Moving Frames , 2002 .

[7]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[8]  A. Isihara,et al.  Theory of Liquid Crystals , 1972 .

[9]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[10]  朴 俊植,et al.  On harmonic maps , 1989 .

[11]  S. Hess,et al.  Alignment tensor versus director: Description of defects in nematic liquid crystals. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  Karen Uhlenbeck,et al.  Boundary regularity and the Dirichlet problem for harmonic maps , 1983 .

[13]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[14]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[15]  Fanghua Lin,et al.  Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents , 1999 .

[16]  Apala Majumdar,et al.  Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond , 2008, 0812.3131.

[17]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[18]  F. Lin,et al.  Stability of singularities of minimizing harmonic maps , 1989 .

[19]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[20]  E. Gartland,et al.  INSTABILITY OF RADIAL HEDGEHOG CONFIGURATIONS IN NEMATIC LIQUID CRYSTALS UNDER LANDAU-DE GENNES FREE-ENERGY MODELS , 1999 .

[21]  Giandomenico Orlandi,et al.  Asymptotics for the Ginzburg–Landau Equation in Arbitrary Dimensions , 2001 .

[22]  Michael Frazier,et al.  Studies in Advanced Mathematics , 2004 .

[23]  F. Lin,et al.  Static and dynamic theories of liquid crystals , 2001 .

[24]  E. Gartland,et al.  Fine structure of defects in radial nematic droplets , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[26]  F. C. Frank,et al.  I. Liquid crystals. On the theory of liquid crystals , 1958 .

[27]  J. Nehring,et al.  On the schlieren texture in nematic and smectic liquid crystals , 1972 .

[28]  Epifanio G. Virga,et al.  Variational Theories for Liquid Crystals , 2018 .

[29]  E. Lieb,et al.  Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds , 1988 .

[30]  P. Gennes,et al.  The physics of liquid crystals , 1974 .

[31]  M. Yoneya,et al.  Physics of Liquid Crystals , 2014 .

[32]  Arghir Zarnescu,et al.  Orientability and Energy Minimization in Liquid Crystal Models , 2010, 1009.2688.

[33]  Apala Majumdar,et al.  Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory , 2008, European Journal of Applied Mathematics.

[34]  J. Ericksen.  Liquid crystals with variable degree of orientation , 1991 .

[35]  Michael Struwe.  Uniqueness of harmonic maps with small energy , 1998 .

[36]  L. Simon.  Regularity Theory for Harmonic Maps , 1996 .