In this part we shall discuss the regularity of boundaries of those sets of finite perimeter which arise as minimizers in some of the variational problems considered so far. The following theorem exemplifies the kind of result we shall obtain. We recall from Section 16.2 that E is a local perimeter minimizer (at scale r0) in some open set A, if spt μE = E (recall Remark 16.11) and
whenever EΔF ⊂⊂ B(x, r0) ∩ A and x ∈ A.
TheoremIf n ≥ 2, A is an open set in ℝn, and E is a local perimeter minimizer in A, then A ∩ E is an analytic hypersurface with vanishing mean curvature which is relatively open in A ∩ E, while thesingular set of E in A,
satisfies the following properties:
(i) if 2 ≤ n ≤ 7, then Σ(E; A) is empty;
(ii) if n = 8, then Σ(E; A) has no accumulation points in A;
(iii) if n ≥ 9, then Hs(Σ(E; A)) = 0 for every s > n − 8.
These assertions are sharp: there exists a perimeter minimizer E in ℝ8such that H0(Σ(E; ℝ8)) = 1; moreover, if n ≥ 9, then there exists a perimeter minimizer E in ℝn such that Hn−8(Σ(E;ℝn)) = ∞.
The proof of this deep theorem, which will take all of Part III, is essentially divided into two parts. The first one concerns the regularity of the reduced boundary in A and, precisely, it consists of proving that the locally Hn−1- rectifiable set A ∩ E is, in fact, a C1,γ-hypersurface for every γ ∈ (0, 1).