We prove that B2-convexity
is sufficient for lower semicontinuity of surface energy of partitions of ℝn, 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.
