Regularity results for eikonal-type equations with nonsmooth coefficients

Solutions of the Hamilton–Jacobi equation H(x,−Du(x)) = 1, where H(·, p) is Hölder continuous and the level-sets {H(x, ·) ≤ 1} are convex and satisfy positive lower and upper curvature bounds, are shown to be locally semiconcave with a power-like modulus. An essential step of the proof is the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal C}^{1,\alpha}}$$\end{document} -regularity of the extremal trajectories associated with the multifunction generated by DpH.