A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals

In this short note we prove that the functional I : W-1,W-p(J; R) -> R defined by I(u) = integral(J) integral(J) W(u'(x), u'(y)) dx dy is sequentially weakly lower semicontinuous in W-1,W-p (J, R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull W-sc.