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 second order ordinary differential equations with possibly discontinuous non-monotone nonlinearities. Then, exploiting this tool, some optimal regularity results, recently established by López-Gómez and Omari in [15] , [16] [17] bounded variation solutions non-autonomous quasilinear driven one-dimensional curvature operator, are substantially improved admitting general prescribed curvatures incorporating boundary conditions. The novel approach developed here yields new, deeper, interpretation assumptions introduced our previous papers, simultaneously clarifying their meaning making fully transparent connection principle.