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.

Local simplicity, topology, and sets of finite perimeter

“…Because the area integrand is strictly convex, if we replace a unit ball in R n by a half-ball (by intersecting the ball with a half-space passing through its center), the surface area must decrease. Thus, the surface area change α(n−1)−(1/2)nα(n) must be negative, and this implies the rightmost inequality of (18). 2 We can now establish the local simplicity of the sets E y = {u > y}, for L 1 TV-minimizers u.…”

B2-convexity implies strong and weak lower semicontinuity of partitions of ℝn

“…This surface energy functional was first rigorously considered by F. Almgren [1976] for the special case φ uv = c uv φ for constants c uv satisfying additional hypotheses and for a fixed norm φ satisfying additional regularity hypotheses. It has subsequently been considered more generally; see, for example, [Almgren et al 1993;Ambrosio and Braides 1990a;1990b;Ambrosio et al 2000;Bellettini et al 2006;Braides 1998;Caraballo 1997;2008;Morgan 1997;White 1996]. Although Almgren's restrictions on the functions φ uv were sufficient for lower semicontinuity of the surface energy functional (1) with respect to strong convergence (i.e., convergence in volume of each of the regions separately), his hypotheses were far from necessary [Caraballo 2008].…”

“…It has subsequently been considered more generally; see, for example, [Almgren et al 1993;Ambrosio and Braides 1990a;1990b;Ambrosio et al 2000;Bellettini et al 2006;Braides 1998;Caraballo 1997;2008;Morgan 1997;White 1996]. Although Almgren's restrictions on the functions φ uv were sufficient for lower semicontinuity of the surface energy functional (1) with respect to strong convergence (i.e., convergence in volume of each of the regions separately), his hypotheses were far from necessary [Caraballo 2008]. L. Ambrosio and A. Braides [1990a;1990b] discovered the first necessary and sufficient conditions that the functions φ uv must satisfy for strong lower semicontinuity, an integral condition they named BV-ellipticity.…”

“…As a result, many other conditions on the integrands φ uv have been introduced and studied, such as (B)-convexity (introduced in [Ambrosio and Braides 1990b]; cf. [Ambrosio et al 2000]), joint convexity (see [Ambrosio et al 2000] but also [Ambrosio and Braides 1990b]), LSC1 and LSC3 (introduced in [Caraballo 1997]), B2-convexity (introduced in [Morgan 1997]), and A-convexity, A2-convexity, and directional control (introduced in [Caraballo 2008]). These other conditions are easier to work with than BV-ellipticity, and yet are less restrictive than Almgren's conditions.…”

“…F. Morgan's B2-convexity is particularly important, as it is quite general and relatively easy to work with. In [Caraballo 2008], we showed that Almgren's condition, (B)-convexity, LSC1, LSC3, A-convexity, A2-convexity, and directional control each imply B2-convexity. When those results are combined with the main results of this paper, Theorems 14 and 15, we conclude that each of those conditions on the integrands φ uv is sufficient for both strong and weak lower semicontinuity of the surface energy functional (1).…”