On the extension of the solutions of Hamilton–Jacobi equations Original Research Article

We consider the viscosity solution of a homogeneous Dirichlet problem for the eikonal equation in a bounded set ΩΩ. We suppose that the Hamiltonian, H(x,p)=〈A(x)p,p〉−1H(x,p)=〈A(x)p,p〉−1, is strictly convex w.r.t. the variables pp and of class C1,1C1,1 w.r.t. the variables xx. Then the solution of the Dirichlet problem admits an extension to a neighbourhood of ΩΩ, View the MathML sourceu¯, such that View the MathML sourceu¯ is still a viscosity solution of the eikonal equation if and only if ∂Ω∂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.