Regularity Results for Eikonal-Type Equations with Nonsmooth Coefficients

Abstract

Solutions of the Hamilton-Jacobi equation $H(x,-Du(x))=1$, with $H(\cdot,p)$ Hölder continuous and $H(x,\cdot)$ convex and positively homogeneous of degree 1, are shown to be locally semiconcave with a power-like modulus. An essential step of the proof is the ${\mathcal C}^{1,α}$-regularity of the extremal trajectories associated with the multifunction generated by $D_pH$.