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On blow-ups of sets with finite fractional variation

Giorgio Stefani

TL;DR

This work investigates the infinitesimal geometry of sets with locally finite fractional α-variation by coupling the fractional BV framework with tangent-measure theory. Using a Preiss–Mattila style analysis of tangent measures and their iterates, the authors derive a rigidity result: for $|D^α 1_E|$-almost every point on the fractional reduced boundary $\mathscr F^α E$, any nontrivial tangent with locally finite integer perimeter must be a half-space oriented by the fractional inner normal $ν_E^α(x)$. This extends De Giorgi-type blow-up phenomena to the fractional setting and provides a precise infinitesimal model for the geometry of fractional perimeters. The results illuminate the structure of tangent sets in fractional BV theory and suggest robust rigidity under mild fractional BV hypotheses, with potential implications for rectifiability and fractional perimeter analysis.

Abstract

Given $α\in(0,1)$ and a set $E\subset\mathbb{R}^N$ with locally finite fractional $α$-variation, we show that for $|D^α\mathbf 1_E|$-a.e. $x$, every non-trivial tangent set of $E$ at $x$ with locally finite integer perimeter is a half-space oriented by the fractional inner unit normal of $E$ at $x$.

On blow-ups of sets with finite fractional variation

TL;DR

This work investigates the infinitesimal geometry of sets with locally finite fractional α-variation by coupling the fractional BV framework with tangent-measure theory. Using a Preiss–Mattila style analysis of tangent measures and their iterates, the authors derive a rigidity result: for -almost every point on the fractional reduced boundary , any nontrivial tangent with locally finite integer perimeter must be a half-space oriented by the fractional inner normal . This extends De Giorgi-type blow-up phenomena to the fractional setting and provides a precise infinitesimal model for the geometry of fractional perimeters. The results illuminate the structure of tangent sets in fractional BV theory and suggest robust rigidity under mild fractional BV hypotheses, with potential implications for rectifiability and fractional perimeter analysis.

Abstract

Given and a set with locally finite fractional -variation, we show that for -a.e. , every non-trivial tangent set of at with locally finite integer perimeter is a half-space oriented by the fractional inner unit normal of at .

Paper Structure

This paper contains 12 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Fractional variation
  4. Blow-up Theorem
  5. Main result
  6. Preliminaries
  7. Radon measure
  8. Tangent measures
  9. Proof of the main result
  10. Properties of tangent sets
  11. Iterated tangent sets
  12. Proof of Theorem 1.2

Key Result

Theorem 1.1

Let $\mathbf 1_E\in BV^{\alpha,\infty}_{\rm loc}(\mathbb{R}^N)$ and $x\in\mathscr{F}^\alpha E$. Then, $\mathop{\mathrm{Tan}}\nolimits(E,x)\ne\emptyset$ and any $F\in\mathop{\mathrm{Tan}}\nolimits(E,x)$ is such that $\mathbf 1_F\in BV^{\alpha,\infty}_{\rm loc}(\mathbb{R}^N)$ with $\nu_F^\alpha=\nu_E^

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Corollary 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of \ref{['res:preissset']}
  • Theorem 3.3
  • proof
  • proof : Proof of \ref{['res:main']}