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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) in the interior, we prove full regularity up to the boundary for the minimizers. As a consequence, in a relevant range (which we call the Lyuksyutov regime) of parameters of the model we show that even without the norm constraint isotropic melting is anyway avoided in the energy minimizing configurations. Finally, we describe a class of boundary data including radial anchoring which yield in both the previous situations as minimizers smooth configurations whose level sets of the biaxiality carry nontrivial topology. Results in this paper will be largely employed and refined in the next papers of our series. In particular, in [DMP2], we will prove that for smooth minimizers in a restricted class of axially symmetric configurations, the level sets of the biaxiality are generically finite union of tori of revolution.
We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axisymmetric domains and in a restricted class of $mathbb{S}^1$-equivariant (i.e., axially symmetric) configurations. We
In this paper, we study the connection between the Ericksen-Leslie equations and the Beris-Edwards equations in dimension two. It is shown that the weak solutions to the Beris-Edwards equations converge to the one to the Ericksen-Leslie equations as
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Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains or in $mathbb{R}^2$. We present a purely Lagrangian proof that the solution map is strongly continuous in $L^p$ for all $pin [1