Neumann-Type Boundary Conditions for Hamilton-Jacobi Equations in Smooth Domains

Neumann or oblique derivative boundary conditions for viscosity solutions of Hamilton-Jacobi equations are considered. As developed by P.L. Lions, such boundary conditions are naturally associated with optimal control problems for which the state equations employ "Skorokhod" or reflection dynamics to ensure that the state remains in a prescribed set, assumed here to have a smooth boundary. We develop connections between the standard formulation of viscosity boundary conditions and an alternative formulation using a naturally occurring discontinuous Hamiltonian which incorporates the reflection dynamics directly. (This avoids the dependence of such equivalence on existence and uniqueness results, which may not be available in some applications.) At points of differentiability, equivalent conditions for the boundary conditions are given in terms of the Hamiltonian and the geometry of the state trajectories using optimal controls.