A necessary and sufficient condition for lower semicontinuity. (June 2015) - Kristensen, Jan
Abstract: It is well-known that W 1, p quasiconvexity is a necessary condition for sequential weak lower semicontinuity of the variational integral I [ u, Ω ] = ∫ Ω F ( ∇ u ( x ) ) d x on the Sobolev space W 1, p, and that it is sufficient too provided that the integrand F satisfies suitable growth conditions related to the exponent p . 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.