First variation of anisotropic energies and crystalline mean curvature for partitions

We rigorously derive the notion of crystalline mean curvature of an anisotropic partition with no restriction on the space dimension. Our results cover the case of crystalline networks in two dimensions, polyhedral partitions in three dimensions, and generic anisotropic partitions for smooth anisotropies. The natural equilibrium conditions on the singular set of the partition are derived. We discuss several examples in two dimensions (also for two adjacent triple junctions) and one example in three dimensions when the Wulff shape is the unit cube. In the examples we also analyze the stability of the partitions.