SINGULARITIES OF SOLUTIONS OF HAMILTON-JACOBI EQUATIONS

The paper’s Theorem 5.6 proves that the intrinsic singular characteristic y(t)=argmax_y{u(y)−A_t(x,y)} is Lipschitz for small time and that, for smooth approximants u_m, the corresponding maximizers y_m converge uniformly to y on compact subintervals. The candidate solution reproduces the same construction (localization/uniqueness via small-time strong convexity of A_t, C^2-approximation u_m with uniform bounds, stability of minimizers, and an implicit-function argument to obtain an equi-Lipschitz family y_m, then pass to the limit). Minor differences in constants and one optional estimate (a bound on ∂_{ty}^2A_t) do not affect correctness. Overall, the reasoning aligns with the paper’s argument and yields the same conclusion.