Crystals and Polycrystals in ℝ n : Lower Semicontinuity and Existence

Abstract

We give a simple, new algebraic condition, directional control, which is sufficient for lower semicontinuity of surface energy and which is also very easy to check in practice, and we discuss and relate several other sufficient conditions. We establish an existence theorem for surface energy minimizers. We also show how to apply these results to minimal partitions, immiscible fluids (with and without gravity), soap bubble clusters, and curvature flow of polycrystals. In some cases, we use our results to give short, alternative proofs of important existence results in the literature. Our techniques are representative of those which could be used for many variational problems, both static and dynamic, involving interfaces. Our setting is that of the sets of finite perimeter and integral currents of geometric measure theory.