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The $\eps-\eps^β$ property for clusters with double density

A. Pratelli, V. Scattaglia

Abstract

This article is devoted to extend the "$\eps-\eps^β$ property" to the case of clusters in an Euclidean space with a double density.

The $\eps-\eps^β$ property for clusters with double density

Abstract

This article is devoted to extend the " property" to the case of clusters in an Euclidean space with a double density.

Paper Structure

This paper contains 5 sections, 10 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Basic notation and properties of sets of finite perimeter and of clusters
  3. First step: the case $\alpha=0$
  4. Second step: the Infiltration Lemma
  5. Proof of the main result

Key Result

Theorem 1.1

Let us assume that $f:\mathbb R^N\to (0,+\infty)$ and $g:\mathbb R^N\times \mathbb S^{N-1}\to (0,+\infty)$ are two l.s.c. and locally bounded functions, and that $g$ is locally $\alpha$-Hölder in the first variable for some $0\leq \alpha\leq 1$. Let moreover $\mathcal{E}$ be a $m$-cluster of finite In addition, if $\alpha=0$ (thus $\beta=\frac{N-1}{N}$) and $g$ is continuous in the first variable

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: Vol'pert Theorem
  • Theorem 1.5: Blow-up Theorem
  • Definition 1.6: The $\varepsilon-\varepsilon^\beta$ property for clusters
  • Remark 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 14 more