Sufficient conditions for the existence of viscosity solutions for nonconvex Hamiltonians

We study a sufficient geometric condition for the existence of a $W^{1,\infty}(\Omega)$ viscosity solution of the Hamilton--Jacobi equation $$ \left\{ \begin{array}{@{}r@{\;}c@{\;}lcc} F(Du) & = & 0 & \mbox{in} & \Omega, \\[2pt] u & = & \varphi & \mbox{on} & \partial\Omega, \end{array} \right. $$ where $\Omega \subset {\mathbb R}^n$ and $F:{\mathbb R}^n\to {\mathbb R}$ are not necessarily convex.