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TORUS-LIKE SOLUTIONS FOR THE LANDAU-DE GENNES MODEL. PART I: THE LYUKSYUTOV REGIME

Abstract : We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a corresponding physically relevant norm constraint (Lyuksyutov constraint), we prove full regularity up to the boundary for the (constrained) minimizers. As a consequence, in a relevant range of parameters (which we call Lyuksyutov regime), we show that unconstrained minimizers do not exhibit isotropic melting. In the case of a nematic droplet, the radial hedgehog is even shown to be an unstable equilibrium. Finally, we describe a class of boundary data including radial anchoring for which constrained or unconstrained minimizers are smooth configurations whose biaxiality level sets carry nontrivial topology. Results of this paper will be largely employed and refined in the next of our series. In particular in [16], where we prove that biaxiality level sets are generically finite unions of tori for smooth equilibrium configurations minimizing the energy in a restricted axially symmetric class.
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Submitted on : Thursday, October 29, 2020 - 4:25:24 PM
Last modification on : Tuesday, November 3, 2020 - 3:29:31 AM

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Federico Dipasquale, Vincent Millot, Adriano Pisante. TORUS-LIKE SOLUTIONS FOR THE LANDAU-DE GENNES MODEL. PART I: THE LYUKSYUTOV REGIME. Archive for Rational Mechanics and Analysis, Springer Verlag, In press, ⟨10.1007/s00205-020-01582-8⟩. ⟨hal-02458774v2⟩

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