Book contents
- Frontmatter
- Contents
- Preface
- Notation
- PART I RADON MEASURES ON ℝn
- PART II SETS OF FINITE PERIMETER
- PART III REGULARITY THEORY AND ANALYSIS OF SINGULARITIES
- 21 (Λ, r0)-perimeter minimizers
- 22 Excess and the height bound
- 23 The Lipschitz approximation theorem
- 24 The reverse Poincaré inequality
- 25 Harmonic approximation and excess improvement
- 26 Iteration, partial regularity, and singular sets
- 27 Higher regularity theorems
- 28 Analysis of singularities
- PART IV MINIMIZING CLUSTERS
- References
- Index
28 - Analysis of singularities
Published online by Cambridge University Press: 05 October 2012
- Frontmatter
- Contents
- Preface
- Notation
- PART I RADON MEASURES ON ℝn
- PART II SETS OF FINITE PERIMETER
- PART III REGULARITY THEORY AND ANALYSIS OF SINGULARITIES
- 21 (Λ, r0)-perimeter minimizers
- 22 Excess and the height bound
- 23 The Lipschitz approximation theorem
- 24 The reverse Poincaré inequality
- 25 Harmonic approximation and excess improvement
- 26 Iteration, partial regularity, and singular sets
- 27 Higher regularity theorems
- 28 Analysis of singularities
- PART IV MINIMIZING CLUSTERS
- References
- Index
Summary
In Theorem 26.5, we have proved the C1,γ-regularity of the reduced boundaries A∩E 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, then ℋs(Σ(E; A)) = 0 for every s > n − 8.
There exists a perimeter minimizer E in ℝ8with H0(Σ(E;∈8)) = 1. If n ≥ 9, then there exists a perimeter minimizer E in ℝn 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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- Information
- Sets of Finite Perimeter and Geometric Variational ProblemsAn Introduction to Geometric Measure Theory, pp. 362 - 390Publisher: Cambridge University PressPrint publication year: 2012