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  • Print publication year: 2012
  • Online publication date: October 2012

28 - Analysis of singularities

Summary

In Theorem 26.5, we have proved the C1,γ-regularity of the reduced boundaries AE of any (Λ, r0)-perimeter minimizers E in some open set A. This chapter is devoted to the study of the singular set

aiming to provide estimates on its possible size. From the density estimates of Theorem 21.11, we already know that ℋn−1(Σ(E; A)) = 0. This result can be largely strengthened, as explained in the following theorem.

Theorem 28.1 (Dimensional estimates of singular sets of (Λ, r0)-perimeter minimizers) If E is a (Λ, r0)-perimeter minimizer in the open set A ⊂ ℝn, n ≥ 2, with Λr0 ≤ 1, then the following statements hold true:

(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, thens(Σ(E; A)) = 0 for every s > n − 8.

There exists a perimeter minimizer E in8with H0(Σ(E;∈8)) = 1. If n ≥ 9, then there exists a perimeter minimizer E inn with Hn−8(Σ(E;ℝn)) = ∞.

Let us now sketch the proof of this deep theorem. We start by looking at the blow-ups Ex, r of a (Λ, r0)-minimizer E at a singular point x. In Theorem 28.6, we show that, loosely speaking, the Ex, r converge to a cone K with vertex at 0.

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