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 Δ D 1 be the Dirichlet heat evolution of the constant initial temperature. We prove that if the exterior normal derivative ∂ ν u ( ⋅ , t j ) is constant on ∂Ω along a sequence t j ↓ 0 , then Ω is a ball, assuming only ∂ Ω ∈ C 2 . 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 ∂ ν u ( y , t ) = − 1 π t + 1 2 H ( y ) + 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- C 1 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.