B2-convexity and surface energy of space partitions

Abstract

We prove that B2-convexity is not equivalent to lower semicontinuity of surface energy of partitions of R m , for any m ⩾ 2 . B2-convexity, formulated by F. Morgan in 1995, is an extension of F. Almgren's partitioning regularity, L. Ambrosio and A. Braides's ( B)-convexity, the author's LSC1 condition, and several other convexity-type conditions. It is equivalent to BV-ellipticity, and hence to lower semicontinuity, in important special cases, as with immiscible fluids or soap bubble clusters. The question of whether B2-convexity is necessary for lower semicontinuity of surface energy in general has been open since the condition was first formulated. In addition to settling that question, we establish that several other sufficient conditions from the literature are not necessary for lower semicontinuity. Finally, we show that B2-convexity is not necessary for L. Ambrosio and A. Braides's joint convexity, a useful algebraic condition which might be necessary for lower semicontinuity.