Discrete-Time Heat-Flux Rigidity Under C² Boundary Regularity

Let Ω ⊂ R^N, N ≥ 2, be a bounded connected domain, and let u(t) = e^t∆_D 1 be the Dirichlet heat evolution of the constant initial temperature. We prove that if the exterior normal derivative ∂_νu(·,t_j) is constant on ∂Ω along a sequence t_j ↓ 0, then Ω is a ball, assuming only ∂Ω ∈ C². This answers, in the classical curvature regime, the finite-regularity question posed by Cavallina and Pinamonti for their short-time discrete-flux theorem. The key result is the uniform expansion ∂_νu(y,t) = −1(πt)^-½ + H(y)/2 + o(1) as t ↓ 0, where H is the sum of the principal curvatures with respect to the inward unit normal. More quantitatively, the remainder is bounded by C(√t + ω_H(C√t)) , with ω_H the modulus of continuity of H. If ∂Ω ∈ C^2,α, this gives the rate O(t^α/2). The proof uses a boundary-layer parametrix, ambient Euclidean mollification of a normal extension of H, and an L^∞-to-C¹ estimate for the Dirichlet heat semigroup. The ambient regularization avoids differentiating the metrics of parallel hypersurfaces and therefore does not require third derivatives of the boundary