Effective fronts of polygon shapes in two dimensions

We study the effective fronts of first order front propagations in two dimensions (n=2) in the periodic setting. Using PDE-based approaches, we show that for every alpha is an element of(0,1), the class of centrally symmetric polygons with rational vertices (i.e., vectors in boolean OR(lambda is an element of R)lambda Z(2)) and nonempty interior is admissible as effective fronts for front speeds in C-1,C-alpha(T-2, (0, infinity)). This result can also be formulated in the language of stable norms corresponding to periodic metrics in T-2. Similar results were known long ago when n >= 3 for front speeds in C-infinity(T-n,(0, infinity)). The two-dimensional case is much more subtle due to topological restrictions. In fact, for given C-1,C-1(T-2,(0, infinity)) front speeds, the effective front is C-1 and hence cannot be a polygon. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.