on the extension of the solutions of hamilton–jacobi equations

we consider the viscosity solution of a homogeneous dirichlet problem for the eikonal equation in a bounded set ω. we suppose that the hamiltonian, h(x, p) = ⟨a(x)p, p⟩ − 1, is strictly convex w.r.t. the variables p and of class c^1,1 w.r.t. the variables x. then the solution of the dirichlet problem admits an extension to a neighbourhood of ω, u, such that u is still a viscosity solution of the eikonal equation if and only if ∂ω satisfies an exterior sphere condition. the above result, in particular, provides a characterization of the boundary singularities and a regularity theorem (up to the boundary) for the solution of the eikonal equation.