On Variational Aspects of the Theory of Liquid Crystals with Variable Degree of Orientation
Let us consider a nematic liquid crystal occupying in the three dimensional Euclidean space R 3 a bounded region Ω with Lipschitz continuous boundary ∂Ω. We put ourselves into the framework of Ericksen’s theory of nematic liquid crystals with variable degree of orientation (, see also , p. 331) and we assume that the crystal is determined by the orientation vector n : Ω → R 3 with n = 1 and the degree of orientation s : Ω → [-1/2, 1 ]. The Prank energy functional F E in a setting of the one constant approximation is given by where ∇ = (∂ j ) stands for the gradient operator, k E and k are positive module, σ0 : Ω → R is De Gennes potential of the form with positive constants a, b, c and d.
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