Appl Math Optim 53:359–381 (2006)
2006 Springer Science+Business Media, Inc.
Neumann-Type Boundary Conditions for Hamilton–Jacobi
Equations in Smooth Domains
Martin V. Day
Department of Mathematics, Virginia Tech,
Blacksburg, VA 24061-0123, USA
Abstract. Neumann or oblique derivative boundary conditions for viscosity so-
lutions 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 con-
ditions for the boundary conditions are given in terms of the Hamiltonian and the
geometry of the state trajectories using optimal controls.
Key Words. Viscosity solution, Boundary condition, Skorokhod problem.
AMS Classiﬁcation. 35F30, 49L25.
Lions developed the notion of Neumann-type boundary conditions for viscosity solutions
of Hamilton–Jacobi equations in . The now standard expression of these is given in (16)
below. He showed that this provided the appropriate notion of solution for the Hamilton–
Jacobi equations describing the value functions of control problems or differential
This research was supported in part by NSF Grant DMS-0102266.