A necessary and sufficient condition for lower semicontinuity

It is well-known that W1,p quasiconvexity is a necessary condition for sequential weak lower semicontinuity of the variational integral I[u,Ω]=∫ΩF(∇u(x))dx on the Sobolev space W1,p, and that it is sufficient too provided that the integrand FF satisfies suitable growth conditions related to the exponent pp. We show that for extended real-valued integrands a closely related convexity condition defined in terms of gradient Young measures is both necessary and sufficient for lower semicontinuity in the more general–and flexible–setting of compensated compactness. Our main results identify the relaxation (lower semicontinuous envelope) in two related situations.