On passing to the limit for non-convex variational problems

Abstract

In this paper, we consider a sequence of integral functionals F_n:X→(−∞,+∞], where X is the set of those functions u belonging to W^1,p(0, T), p > 1, satisfying: u(0) = A, u(T) = B. For every n∈N,F_n is represented by the sum of two integrands, where the first one is T-periodic in time and non-convex with respect to u′ and the second one depends only on u.

We give a necessary and sufficient condition in order to obtain the existence of an integral functional F_∞:X→(−∞,+∞] such that, for every minimizing sequence (u_n) converging to u_∞, the lower limit of the corresponding sequence F_n(u_n) coincides with F_∞(u_∞). The integrand function in F_∞ does not depend on time and, in general, it is non-convex with respect to u′.