Plane-like minimizers and di erentiability of the stable norm

In this paper we investigate the strict convexity and the di erentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always di erentiable in totally irrational directions, while in other directions, it is di erentiable if and only if the corresponding plane-like minimizers satisfying a strong Birkho property foliate the torus. We also discuss the issue of the uniqueness of the correctors for the corresponding homogenization problem.