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 $\alpha\in (0,1)$, the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts for given front speeds in $C^{1,\alpha}(\mathbb T^2,(0,\infty))$. This result can also be formulated in the language of stable norms corresponding to periodic metrics in $\mathbb T^2$. Similar results were known long time ago when $n\geq 3$ for front speeds in $C^{\infty}(\mathbb T^n,(0,\infty))$. Due to topological restrictions, the two dimensional case is much more subtle. In fact, the effective front is $C^1$, which cannot be a polygon, for given $C^{1,1}(\mathbb T^2,(0,\infty))$ front speeds. 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.