We prove local optimality principles for viscosity super and subsolutions of Hamilton-Jacobi equations with unbounded ingredients. We apply these results to characterize the (possibly multiple) discontinuous solutions of mixed Dirichlet and constrained boundary value problems.
"Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints." Differential Integral Equations 12 (2) 275 - 293, 1999. https://doi.org/10.57262/die/1367265632