Weak compactness of Fréchet-derivatives: application to composition operators

摘要

We prove a result on compactness properties of Fr??chet-derivatives which implies that the Fr??chet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.