Representations of solutions of Hamilton-Jacobi equations

In this paper we report on some classical and more recent results about representation formulas for generalized solutions of the evolution partial differential equation ut + H(x,Du) = 0 , (x, t) 2 IRN × (0,+1) (1.1) We consider here only the case where H = H(x, p) is a convex function with respect to the p variable. In this setting, representation formulas can be obtained by exploiting the well - known connection existing via convex duality between the Hamilton - Jacobi equation (1.1) with Calculus of Variations or, more generally, Optimal Control problems