Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations

We consider a class of non convex scalar functionals of the form F(u) = ∫Ω f(x,u,Du)dx, under standard assumptions of regularity of the solutions of the associated relaxed problem and of local affinity of the bipolar f ** of f on the set {f ** < f}. We provide an existence theorem, which extends known results to lagrangians depending explicitly on the three variables, by the introduction of integro-extremal minimizers of the relaxed functional which solve the equation f** (x, u,Du) - f(x, u,Du) =0, or the opposite one, almost everywhere and in viscosity sense. © 2007 Springer-Verlag.