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

TL;DR

This paper develops a refined strong maximum principle for certain second order ODEs and uses it to improve regularity results for solutions of one-dimensional curvature equations with general boundary conditions.

Contribution

It introduces a new, deeper interpretation of assumptions in curvature equations by leveraging an enhanced maximum principle, broadening the scope of regularity results.

Findings

Refined maximum principle for second order ODEs with discontinuous nonlinearities

Improved regularity results for solutions with general prescribed curvatures

Clarification of assumptions' meaning in curvature equations

Abstract

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 Lopez-Gomez 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 new 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.