I am currently interested in viscosity solutions of the first-order Hamilton-Jacobi equations under Neumann boundary conditions. For example, I consider the following initial boundary value problem:
$ \begin{cases} u_t(x,t)+H(x,t,Du(x,t))=0 & \mathrm{in} \ \Omega \times (0,T), \\ \frac{\partial u}{\partial n}(x,t)=0 & \mathrm{in} \ \partial \Omega \times (0,T), \\ u(x,0)=u_0(x) & \mathrm{in} \ \Omega. \end{cases} $
I would like to know some representation formulas for the solution of this equation, but I do not know which references to consult. If you know of any, please let me know.