Abstract
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 \({{\mathcal C}^{1,\alpha}}\) -regularity of the extremal trajectories associated with the multifunction generated by D p H.
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Cannarsa, P., Cardaliaguet, P. Regularity results for eikonal-type equations with nonsmooth coefficients. Nonlinear Differ. Equ. Appl. 19, 751–769 (2012). https://doi.org/10.1007/s00030-011-0150-1
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DOI: https://doi.org/10.1007/s00030-011-0150-1