B2-convexity implies strong and weak lower semicontinuity of partitions of ℝn

Abstract

We prove that B2-convexity is sufficient for lower semicontinuity of surface energy of partitions of R n , for any n 2. We establish lower semicontinuity in the usual strong topology, assuming the regions converge in volume. We also establish lower semicontinuity in the more general situation in which we suppose integral currents associated with individual regions converge to some integral current in the weak topology of integral currents. B2-convexity, formulated by F. Morgan in 1995, is a powerful condition since it is easy to work with and since many other conditions from the literature imply it. Our results therefore imply that each of those conditions is sufficient for strong and weak lower semicontinuity of surface energy. We establish other results of independent interest, including a Lebesgue point theorem for partitions and a localization theorem, which shows that if lower semicontinuity holds locally then it holds globally.