  This work is concerned with Hamilton-Jacobi equations of evolution type posed
in domains and supplemented with boundary conditions. Hamiltonians are coercive
but are neither convex nor quasiconvex. We analyse boundary conditions when
understood in the sense of viscosity solutions. This analysis is based on the
study of boundary conditions of evolution type. More precisely, we give a new
formula for the relaxed boundary conditions derived by J. Guerand (J. Differ.
Equations, 2017). This new point of view unveils a connection between the
relaxation operator and the classical Godunov flux from the theory of
conservation laws. We apply our methods to two classical boundary value
problems. It is shown that the relaxed Neumann boundary condition is expressed
in terms of Godunov's flux while the relaxed Dirichlet boundary condition
reduces to an obstacle problem at the boundary associated with the lower
non-increasing envelope of the Hamiltonian.
