Existence of surface energy minimizing partitions of ℝⁿ satisfying volume constraints

We give the first proof, with independent smooth norms ϕ i j , \phi _{ij}, of the existence of surface energy minimizing partitions of R n \mathbb {R}^{n} into regions having prescribed volumes. Our existence proof significantly extends that of F. Almgren, who in 1976 gave the first such results for the special case in which each ϕ i j \phi _{ij} is a scalar multiple of a fixed smooth ϕ : \phi : ϕ i j = c i j ϕ \phi _{ij}=c_{ij}\phi . Most materials are polycrystalline and do not have surface energy density functions which are scalar multiples of one another, so it is important to extend the theory by removing this restriction, as we have done. We also discuss connections with polycrystalline evolution problems.