Optimal regularity results for the one-dimensional prescribed curvature equation via the strong maximum principle

A refined version of the strong maximum principle is proven for a class of second order ordinary differential equations with possibly discontinuous non-monotone nonlinearities. Then, exploiting this tool, some optimal regularity results, recently established by L\'opez-G\'omez and Omari for the bounded variation solutions of non-autonomous quasilinear equations driven by the one-dimensional curvature operator, are substantially improved by admitting general prescribed curvatures and by incorporating general boundary conditions. The novel approach developed here yields a new, deeper, interpretation of the assumptions introduced in our previous papers, simultaneously clarifying their meaning and making fully transparent their connection with the strong maximum principle.