Cauchy problems for stationary Hamilton-Jacobi equations under mild regularity assumptions

For a Hamiltonian enjoying rather weak regularity assumptions, we provide necessary and sufficient conditions for the existence of a global viscosity solution to the corresponding stationary Hamilton–Jacobi equation at a fixed level a, taking a prescribed value on a given closed subset of the ground space. The analysis also includes the case where a is the Man ̃ ́e critical value. Our results are based on a metric method extending Maupertuis approach. For general underlying spaces, compact or noncompact, we give a global ver- sion of the classical characteristic method based on the notion of a–characteristic. In the compact case, we propose an inf-sup formula producing the minimal so- lution of the problem, where the generalized Aubry set is involved.