Geometrical exact solutions for confocal lamellar textures yielding confinement and faceting phenomena

We study the equilibrium textures of lamellar bodies in cells limited by two parallel plates interacting only with the orientations of the layers but not with their positions. For such 'loose' anchorings, confocal textures are expected, i.e. textures of equidistant and parallel curved layers. When all defects lie outside the smectic body, with the aid of a fortunate change of variables, we find an exact solution of the nonlinear equilibrium equation describing the textures. Instead of being governed by a differential equation, the equilibrium textures are governed by a finite equation that gives the common evolute f of the layers. The equilibrium texture can then be deduced from f by a geometrical construction. This approach is reminiscent of that introduced by Wulff (1901) in order to determine the equilibrium shape of solid crystals. Our main result is the existence, for certain boundary conditions and according to the shape of the surface energy function, of confined distortions and faceted textures. Faceted textures characterize the presence of an angular point in the surface energy, a feature currently debated.