Regularity Results for Eikonal-Type Equations with Nonsmooth Coefficients

Solutions of the Hamilton-Jacobi equation $H(x,-Du(x))=1$, with $H(\cdot ,p)$ H"older 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 ${\mathcal{C}}^{1,\alpha }$-regularity of the extremal trajectories associated with the multifunction generated by $D_{p}H$.