Viscosity solutions for partial differential equations with Neumann type boundary conditions and some aspects of Aubry–Mather theory

We study partial differential inequalities (PDI) of the type where NK(⋅) is the normal cone to the set K. We prove existence of a constant such that the PDI of Hamilton–Jacobi type has a unique (global) Lipschitz viscosity solution. We provide a formula to calculate this constant. Moreover, we define a subset of K such that any two solutions of the previous PDI which coincide on will coincide on K. Our paper generalizes results of the case without boundary conditions for convex Hamiltonians obtained by L.C. Evans and A. Fathi.