Almost classical solutions of Hamilton-Jacobi equations

We study the existence of everywhere differentiable functions which axe almost everywhere solutions of quite general Hamilton-Jacobi equations on open subsets of ℝ ^d or on d-dimensional manifolds whenever d ≥ 2. In particular, when M is a Riemannian manifold, we prove the existence of a differentiable function u on M which satisfies the Eikonal equation || ▽u(x)|| _x = 1 almost everywhere on M.