Synopsis

A cluster ε in ℝn is a finite disjoint family of sets of finite perimeter with finite and positive Lebesgue measure (note: the chambers ε(h) of ε are not assumed to be connected/indecomposable). By convention, denotes the exterior chamber of ε. The perimeter P(ε) of ε is defined as the total (n − 1)-dimensional Hausdorff measure of the interfaces of the cluster,

Denoting by b(ε) the vector in whose hth entry agrees with ∣ε(h)∣, we shall say that ε is a minimizing cluster in ℝn if spt με(h) = ε(h) for every h = 1,…, N, and, moreover, P(ε) ≤ P(ε′) whenever m(ε′) = m(ε). By a partitioning problem in ℝn, we mean any variational problem of the form

corresponding to the choice of some m. Proving the following theorem will be the main aim of Part IV. The existence and regularity parts will be addressed, respectively, in Chapter 29 and Chapter 30.

Theorem (Almgren's theorem) If n, N ≥ 2 and then there exist minimizers in the partitioning problem defined bym. If ε is an N-minimizing cluster in ℝn, then ε is bounded. If 0 ≤ h ≤ k ≤ N, then ε(h) ∩ ε(k) is an analytic constant mean curvature hypersurface in ℝn, relatively open inside ε(h) ∩ ε(k). Finally,

This existence and almost everywhere regularity theorem is one of the main results contained in the founding work for the theory of minimizing clusters and partitioning problems, that is Almgren's AMS Memoir [Alm76].