This chapter is devoted to the discussion of the regularity properties of minimizing clusters. We shall first prove the following Hn−1-a.e. regularity result in arbitrary dimension (Theorem 30.1); then, in Section 30.3, we shall focus on the planar case, and achieve a more precise result, including a complete description of near-to-singularities behavior; see Theorem 30.7.
Theorem 30.1If ε is a minimizing N-cluster in ℝn and 0 ≤ h ≤ k ≤ N, then ε(h, k) is an analytic hypersurface with constant mean curvature in ℝn, which is relatively open in ε(h) ∩ ε(k), and satisfies
The key tool in the proof of Theorem 30.1 is the infiltration lemma of Section 30.1, which allows us, for example, to prove each chamber of the cluster to be a volume-constrained perimeter minimizer in a neighborhood of its interface points; see Corollary 30.3. In Section 30.2 we shall combine this result with some density estimates in order to complete the proof of (30.1).
The following lemma shows that if some chambers of a minimizing cluster fill most of B(x, 2r), then they completely fill B(x, r).