Just as “nonlinear” is understood in mathematics to mean “non necessarily linear,” we intend the term “nonsmooth” to refer to certain situation in which smoothness (differentiability) of the data is not necessarily postulated.

In the previous chapter we treated continuous convex perturbations of C1-functionals. In this one, we will assume a weakened smoothness assumption on functionals. Namely, we will suppose that our functional is only locally Lipschitz. And we will prove an appropriate MPT for this kind of functionals.

His approach is exactly the one we saw for C1-functionals “minus” the fact that there was no result about the existence of a pseudo-gradient vector field for locally Lipschitz functionals. He succeeds in establishing such a result using the Hahn-Banach theorem and the basic properties of the subdifferential calculus and generalized gradients in the sense of F.H. Clarke. The rest of his procedure then becomes “standard.” The paper [202] of Chang proved to be an invaluable reference when dealing with locally Lipschitz critical point theory.