Crystalline curvature flow of planar networks

Abstract

We consider the evolution of a polycrystalline material with three or more phases, in the presence of an even crystalline anisotropy. We analyze existence, uniqueness, regularity and stability of the flow. In particular, if the flow becomes unstable at a finite time, we prove that an additional segment (or even an arc) at the triple junction may develop in order to decrease the energy and make the flow stable at subsequent times. We discuss some examples of collapsing situations that lead to changes of topology, such as the collision of two triple junctions.

Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies

“…In the isotropic equal energy density case we have, in addition, ς i = 1, i = 1 → I S ; and so E γ (Γ ) reduces to E(Γ ), the surface area of Γ . Following our recent work in [9,12], we will restrict ourselves to anisotropic surface energy densities of the form 15) so that…”

“…surface diffusion and mean curvature flow. For these geometric flows we impose force balance conditions at points where different surfaces meet, or where a surface meets a fixed external boundary; see [44,72,34,15] and the references therein. For example, in the case of mean curvature flow with equal constant surface energy densities this results in a 120 • angle condition at triple junction lines, while a 90 • contact angle holds where a surface meets an external boundary.…”

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].…”

“…Energy minimization problems involving surface energy of partitions -or polycrystals -are central to materials science, physics, biology, computer science, image processing, and other fields. Some such applications include crystal growth, tumor growth, annealing of metals, image segmentation, noise reduction in images, as well as the study of cell structures, immiscible fluids, metal foams, and semiconductors; see, for example, [Almgren 1976;Almgren and Taylor 1996;Ambrosio and Braides 1995;Ambrosio et al 2001;2000;Aubert and Kornprobst 2002;Bellettini et al 2002;2006;Braides 1998;Brook et al 2003;Gurtin 1993;1986;Morgan 1997;Mumford and Shah 1989;Osher and Fedkiw 2001;Sethian 1999;Taylor 1978;White 1996]. Indeed, most materials are polycrystalline.…”

Local simplicity, topology, and sets of finite perimeter

“…Sets of locally finite perimeter, those whose characteristic functions have locally bounded variation, have become a natural setting in which to study many problems involving surfaces and interfaces, in areas such as materials science, fluid mechanics, surface physics, image processing, oncology, and computer vision (see, for example, [1], [2], [3], [5]- [9], [11], [12], [14]- [20], [23], [27], [28], [30], [31], [33], [38]- [40], and the many references cited therein). They are general enough to adequately model complex physical phenomena with singularities, they have useful local approximation properties, and they satisfy vital compactness results which do not hold for classes of sets having smooth boundaries.…”