New variational problems in the statics of liquid crystals

摘要: Liquid crystals are anisotropic fluids that exhibit a preferred direction, possibly different at each place. The direction is the optical axis of material, here we describe by unit vector n. Most liquid present phases as temperature varies. We restrict attention to only one these phases: our development concerns nematic crystals. assume region space occupied crystal given and denote it B. Thus n field B into S2 sphere in translation three-dimensional Euclidean e. Formally, $$n:B \to {S^2},B \subset \varepsilon ,{S^2}: = \left\{ {u \in v\left| u \right| 1} \right\},{\text{ where v: }}\varepsilon {\text{ - $$