Hölder regularity of continuous solutions to balance laws and applications in the Heisenberg group

We prove Hölder regularity of any continuous solution u to a 1-D scalar balance law ut+[f(u)]x=g, when the source term g is bounded and the flux f is nonlinear of order ℓ∈ℕ with ℓ≥2. For example, ℓ=3 if f(u)=u3. Moreover, we prove that at almost every point (t,x), it holds u(t,x+h)−u(t,x)=o(|h|1ℓ) as h→0. Due to Lipschitz regularity along characteristics, this implies that at almost every point (t,x), it holds u(t+k,x+h)−u(t,x)=o((|h|+|k|)1ℓ) as |(h,k)|→0. We apply the results to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group.