Almost classical solutions of Hamilton-Jacobi equations

Abstract

We study the existence of everywhere differentiable functions which are almost everywhere solutions of quite general Hamilton-Jacobi equations on open subsets of ${\mathbb{R}}^d$ or on $d$-dimensional manifolds whenever $d\ge 2$. In particular, when $M$ is a Riemannian manifold, we prove the existence of a differentiable function $u$ on $M$ which satisfies the Eikonal equation $\Vert \nabla u(x){\Vert }_x=1$ almost everywhere on $M$.