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.
We prove that the triangle inequalities in general are not equivalent to lower semi-continuity of surface energy. In 1990, Ambrosio and Braides proved that the two conditions are equivalent when the number of regions, s, is three. They gave an example in the plane showing that the triangle inequalities do not imply lower semi-continuity when s ≥ 6. The cases when s = 4 and s = 5 have remained open. In this paper, we resolve these open questions and show that, in ℝm, the triangle inequalities are sufficient for lower semi-continuity if and only if s = 3.
We establish useful upper bounds for the (n − 1)-area of a level set ρ −1 {r} of a general distance function ρ to an (n − 1)-dimensional compact subset C of ޒ n , in terms of r and the area of C. These bounds nicely complement general isoperimetric inequalities that provide lower bounds for the same area. We allow distance functions induced from asymmetric norms, and prove our results without assuming that C is smooth. Unlike standard upper bounds using Federer's Coarea Formula, which hold only for some values of r and which become arbitrarily large if we restrict r to be contained in sufficiently small intervals, our estimates hold for ᏸ 1 -almost every r > 0. Our main result both extends and improves upon an important result of Almgren, Taylor, and Wang. First, our estimates hold for general distance functions. Second, in the case of ordinary distance functions, our estimates are sharper than theirs. Because our estimates hold for ᏸ 1 -almost every r, we can easily integrate to obtain volume estimates, such as those typically required for Hölder continuity theorems for flows in ޒ n . Indeed, Almgren, Taylor, and Wang used a weaker inequality to establish their main Hölder continuity theorem for curvature-driven flow of the boundary of a single crystal. In that setting, our estimate would lead to a similar result, but with a better coefficient.We also establish several general results about asymmetric norms and their associated distance functions to compact sets. For example, the latter are Lipschitz continuous and have, for ᏸ n -almost every x ∈ ޒ n , gradients with norms bounded a priori from above and below.
In the impressive and seminal paper [5], Fred Almgren, Jean Taylor, and Lihe Wang introduced flat curvature flow in R n , a variational time-discretization scheme for (isotropic or anisotropic) mean curvature flow. Their main result asserts the Hölder continuity, in time, of these flows. This essential estimate requires a boundary regularity result, a uniform lower density ratio bound condition, which they proved for each n 3. Similar estimates for Brownian flows, from important work by N. K. Yip on stochastic mean curvature flow [30], also rely on this pivotal regularity result. In this paper, we complete this analysis for the case n = 2 by establishing the necessary uniform lower density ratio bounds.
We prove that B2-convexity is sufficient for lower semicontinuity of surface energy of partitions of ޒ 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.
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