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 c + H(x, ∂v ∂x ) − NK(x) 0 where NK(·) is the normal cone to the set K.We prove existence of a constant c := ¯c 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 K of K such that any two solutions of the previous PDI which coincide on K 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. © 2007 Elsevier Inc. All rights reserved