On viscosity solutions of certain Hamilton-Jacobi equations: Regularity results and generalized Sard's Theorems

International audienceUnder usual assumptions on the Hamiltonian, we prove that any viscosity solution of the corresponding Hamilton-Jacobi equation on the manifold $M$ is locally semiconcave and $C^{1,1}_{loc}$ outside the closure of its singular set (which is nowhere dense in $M$ ). Moreover, we prove that, under additional assumptions and in low dimension, any viscosity solution of that Hamilton-Jacobi equation satisfies a generalized Sard theorem. In consequence, almost every level set of such a function is a locally Lipschitz hypersurface in M