Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime
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Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime Federico Dipasquale, Vincent Millot & Adriano Pisante Communicated by A. Braides
Abstract We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic boundary data. To achieve this result, we first study minimizers subject to a physically relevant norm constraint (the Lyuksyutov constraint), and show their regularity up to the boundary. From this regularity, we rigorously derive the norm constraint from the asymptotic Lyuksyutov regime. As a consequence, isotropic melting is avoided by unconstrained minimizers in this regime, which then allows us to analyse their biaxiality sets. In the case of a nematic droplet, this also implies that the radial hedgehog is an unstable equilibrium in the same regime of parameters. Technical results of this paper will be largely employed in Dipasquale et al. (Torus-like solutions for the Landau- de Gennes model. Part II: topology of S1 -equivariant minimizers. https://arxiv.org/pdf/2008.13676.pdf; Torus-like solutions for the Landaude Gennes model. Part III: torus solutions vs split solutions (In preparation)), where we prove that biaxiality level sets are generically finite unions of tori for smooth configurations minimizing the energy in restricted classes of axially symmetric maps satisfying a topologically nontrivial boundary condition.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Small Energy Regularity Theory: A Tool Box . . . . . . . . . 2.1. Monotonicity Formulae for Approximable Critical Points 2.2. Reflection Across the Boundary . . . . . . . . . . . . . 2.3. The ε-Regularity Theorem . . . . . . . . . . . . . . . . 2.4. Higher Order Regularity . . . . . . . . . . . . . . . . .
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F. Dipasquale et al. 2.5. Bochner Inequality and Uniform Regularity Estimates . . . . . 3. Regularity of Minimizers Under Norm Constraint . . . . . . . . . 3.1. Monotonicity Formulae . . . . . . . . . . . . . . . . . . . . . 3.2. Compactness of Blow-ups and Smallness of the Scaled Energy 3.3. Full Regularity . . . . . . . . . . . . . . . . . . . . . . . . . 4. LdG-Minimizers in the Lyuksyutov Regime . . . . . . . . . . . . . 4.1. A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Lyuksyutov Regime and Absence of Isotropic Melting . . . . . 4.3. Instability of the Melting Hedgehog . . . . . . . . . . . . . . 5. Topology of Minimizers . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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