Existence of Solutions to Boundary Value Problems for Smectic Liquid Crystals

We prove lower semicontinuity and lower bounds for a Chen-Lubensky energy describing nematic/smectic liquid crystals with physically realistic boundary conditions. The Chen-Lubensky energy captures stable phases of the liquid crystal material, ranging from purely nematic or smectic states to coexisting nematic/smectic states. By including appropriate additional terms, the model includes the effects of applied electric or magnetic fields, and/or electrical self-interactions in the case of polarized liquid crystals. As a consequence of our results, we establish existence of minimizers with weak or strong anchoring of the director field (describing molecular orientation) at the boundary, and Dirichlet or Neumann boundary conditions on the smectic order parameter for the liquid crystal material. ∗Research supported by NSF grant DMS-1109459 †Research supported by NSF grant DMS-1109459 ‡Research supported by NRF of Korea with grant number 2011-0014882