Weak lower semicontinuity and relaxation for a class of non-local functionals

Motivated by a recent interest in models of peridynamics, we survey, gather, and summarize various results, that are scattered throughout the literature, about weak lower semicontinuity of some non-local variational problems in which non-locality is expressed in the form of a double integral. Weak lower semicontinuity translates into separate convexity, though one has to incorporate in such a result a most unusual phrase to stress that weak lower semicontinuity ought to be true for all domains. The most striking fact, however, is that relaxation cannot be written and understood as a functional of the same kind but something more involved needs to be found.