BV-ellipticity and lower semicontinuity of surface energy of Caccioppoli partitions of R~n

摘要

We give a new proof that BV-ellipticity is sufficient for lower semicontinuity of surface energy of Caccioppoli partitions of R~n, for any n≥2, with respect to convergence in volume. BV-ellipticity, introduced by L. Ambrosio and A. Braides two decades ago, is the only condition known to be necessary and sufficient for lower semicontinuity in the context of Caccioppoli partitions of R~n; it is analogous, for this setting, to C.B. Morrey's quasi-convexity. We also show, for the first time, that BV-ellipticity suffices for lower semicontinuity with respect to weaker notions of convergence involving the weak and flat topologies on integral currents.