Dynamic boundary conditions for Hamilton-Jacobi equations

A nonstandard dynamic boundary condition for a Hamilton-Jacobi equation in one space dimension is studied in the context of viscosity solutions. A comparison principle, and hence uniqueness, is proved by consideration of an equivalent notion of viscosity solution for an alternative formulation of the boundary condition. The relationship with a Neumann condition is established. Global existence is obtained by consideration of a related parabolic approximation with a dynamic boundary condition. The problem is motivated by applications in superconductivity and interface evolution.