Differentiability of effective fronts in the continuous setting in two dimensions

We study the effective front associated with first-order front propagations in two dimensions ($n=2$) in the periodic setting with continuous coefficients. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Equivalently, the stable norm associated with a continuous $\mathbb Z^2$-periodic Riemannian metric is differentiable at irrational points. This conclusion was obtained decades ago for $C^2$ metrics. There, the $C^2$ regularity was essential to establish ordering and non-crossing properties of minimal orbits, which no longer exist in the merely continuous situation. To the best of our knowledge, our result provides the first nontrivial property of the effective fronts in the continuous setting, which is the standard assumption in the PDE theory.