The confinement of smectics with a strong anchoring

We examine several geometric features of the confinement of lamellar materials in complex geometries. When the anchoring of a sample at its boundaries (smectic-fluid interfaces or smectic-solid substrates interfaces) is strong enough, the geometric approximation of parallel layers can be extended to the bounding surface of the sample. Depending on the concavity of the interface, a strong planar anchoring is then compatible (or not) with a smectic organization in the bulk. We give the simplest smectic organization which satisfies a planar anchoring everywhere on the interface of axisymmetric inverse SmA droplets and compute its curvature energy. Eventually, the reason is given why the textures of direct and inverse SmA droplets are so much different (as it was first noticed in the pioneering work on SmA of Friedel and Grandjean).