This chapter will be entirely devoted to the proof of the existence of minimizing clusters. Precisely, we shall prove the following theorem.
Theorem 29.1For everymand n ≥ 2, there exists a minimizer in the variational problem
If ε is a minimizer in (29.1), then ε is bounded, that is, for some R > 0,
This proof of this theorem, which presents several beautiful ideas, is rather long and technical. It could be advisable to limit a first reading to Section 29.1, where the basic definitions and remarks concerning clusters are introduced, Section 29.2, where an outline of the proof is presented, and Sections 29.5–29.6, where the technique of volume-fixing variations, of fundamental importance also in Chapter 30, is discussed. Sections 29.3–29.4 contain instead the other tools needed to prove Theorem 29.1, which are finally employed in Section 29.7 to prove the existence of minimizing clusters.
Definitions and basic remarks
An N-cluster ε of ℝn is a finite family of sets of finite perimeter with
The sets ε(h) are called the chambers of ε. When the number N of the chambers of ε is clear from the context, we shall use the term “cluster” in place of “N-cluster”. The exterior chamber of ε is defined as
In particular, is a partition of ℝn (up to a set of null Lebesgue measure) and, necessarily, ∣ε(0)∣ = ∞.