DIFFERENTIABILITY OF EFFECTIVE FRONTS IN THE CONTINUOUS SETTING IN TWO DIMENSIONS

The paper proves in two dimensions that for H(y,p)=a(y)|p| with a∈C(T^2,(0,∞)), the effective front has no irrational flats (equivalently, the dual ball is differentiable at every irrational point), via a PDE/characteristics argument and a reduction to mechanical Hamiltonians; see Theorem 1.1 and its reduction through Theorem 1.2(2) and the identification with the stable norm unit ball Da . The candidate’s Step 1–2 are sound (duality between Ha and the stable norm, and the flats↔non-differentiability correspondence), but Step 3 invokes the 'classical' differentiability of the stable norm at irrational directions without the smoothness assumptions under which that result is known classically. Precisely this extension to merely continuous coefficients is the new content of the paper; for C^2 metrics it was long known, but in the continuous setting the paper provides the proof (and shows (iii) holds) . Hence the model assumes the key step rather than establishing it in the generality of the problem, whereas the paper correctly proves it.