A variational scheme for the evolution of polycrystals by curvature

This paper provides a quite general basis for studying the time evolution of n-dimensional polycrystals, whose interfaces move with surface normal velocity proportional to the weighted mean curvature. Grain growth is an important example of such motion. Grain boundaries in annealing metals (such as pure aluminum) move with velocities proportional to their mean curvatures, and their surface tensions depend on the orientations of the various grains. The mathematical framework developed here is flexible enough to encompass this and many other dynamic processes involving multiple interfaces.; Novel constructions lead to a general existence theorem for curvature-based evolutions of n-dimensional polycrystals. It is the first such result which (i) allows for the presence of multiple regions (representing different materials, different phases of the same material, or crystals of the same phase having different orientations). (ii) gives each interface its own interfacial free energy function, which, in turn, may depend on the orientation of the interface (to allow for anisotropy), (iii) gives each interface its own mobility function (to take bulk properties into account), which also can depend on orientation, and (iv) holds for each {dollar}nge2.{dollar}; Given an initial n-dimensional polycrystal P(0) and {dollar}Delta t = 2sp{lcub}-k{rcub} (kge1),{dollar} discrete evolutions {dollar}Psb{lcub}Delta t{rcub}(t){dollar} are defined, where the polycrystal {dollar}Psb{lcub}Delta t{rcub}(tsb0 + Delta t){dollar} at a time {dollar}Delta t{dollar} later (called an E-minimizer) is chosen so as to minimize a certain surface plus bulk energy sum involving {dollar}Psb{lcub}Delta t{rcub} (tsb0).{dollar} The bulk energy term is carefully constructed so that the resulting motion approximates mean curvature flow. The limit evolution, or "flat flow" P(t), is obtained by letting {dollar}Delta t{dollar} go to zero.; A quite general lower semi-continuity theorem is proven, which, combined with a compactness argument, implies the existence of E-minimizers. The main result asserts the existence and Holder continuity of flat flows under very general conditions. Boundary regularity results for E-minimizers are proven. For example, boundaries of E-minimizers have uniform lower density ratio bounds, consist locally of single interfaces, and are ({dollar}gamma, delta{dollar}) restricted sets. Barriers to flat curvature flows are also constructed.; These constructions work without any assumptions about the differentiability of the surface energy functions for the various interfaces, which is important since many surface energy functions for crystalline materials (e.g., for the ice-water interface) are not differentiable. Singularities are handled naturally, by choosing an appropriate space of surfaces in which to work. In the special case where the polycrystal consists of only one solid in its melt, such flows are precisely the "flat {dollar}Phi{dollar} curvature flows" introduced in (10), which in turn agree with classical flows when the data are smooth and elliptic.