Smectic liquid crystals are a phase of matter, in which the constituent molecules tend to arrange themselves in layers. Experimental evidence suggests that the orientation of the smectic layers may or may not vary in a smooth way across the sample; under some circumstances, there might be surface defects, i.e. localised regions of sharp change in the orientation of the layers. In this talk, we discuss a free-discontinuity variational problem for smectic liquid crystals in two dimensions, set in De Giorgi and Ambrosio's space of SBV maps (i.e., special maps of bounded variation). We focus on a specific form of the energy functional, which penalises dislocations of the layers along the jump and still retains some desirable mathematical properties (such as BV-ellipticity), so as to guarantee that minimisers exist. The talk is based on joint work with John M. Ball (Heriot-Watt University, Edinburgh and Hong Kong Institute of Advanced Studies) and Bianca Stroffolini (Università Federico II, Napoli).
