Geometric restrictions for the existence of viscosity solutions

We study the Hamilton-Jacobi equation [GRAPHICS] where F : R-N --> R is not necessarily convex. When Omega is a convex set, under technical assumptions our first main result gives a necessary and suf ficient condition on the geometry of Omega and on D phi for (0.1) to admit a Lipschitz viscosity; solution. When we drop the convexity assumption on O mega, and relax technical assumptions our second main result uses the viabi lity theory to give a necessary condition on the geometry of Omega and on D phi for (0.1) to admit a Lipschitz viscosity solution. (C) Elsevier, Paris .