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On sets with finite distributional fractional perimeter

Giovanni E. Comi, Giorgio Stefani

TL;DR

The paper advances the theory of sets with locally finite distributional fractional perimeter by sharpening blow-up descriptions, establishing a fractional Leibniz rule for intersections with finite fractional perimeter sets, and clarifying the structure of non-local boundaries. It develops a detailed framework using $BV^{\alpha,\infty}_{\mathrm{loc}}$ spaces, fractional gradients $\nabla^{\alpha}$, and non-local gradients $\nabla^{\alpha}_{\mathrm{NL}}$, proving precise decomposition formulas for $D^{\alpha}$ under intersections and products, together with tight control via the local fractional perimeter $P_{\alpha}$. A refined blow-up analysis shows that tangent sets at fractional reduced boundary points are half-space–type in the limit, with a 1D slice $M$ controlling the structure, and introduces the effective fractional reduced boundary $\mathscr{F}^{\alpha}_{e}E$, which aligns with classical boundary notions under appropriate regularity. Finally, the work clarifies the support and nature of the singular part of the fractional variation, relating it to $\partial^-E$ and establishing consequences for the geometry of non-local boundaries, including Ball-type explicit computations of the nonlocal gradient on balls.

Abstract

We continue the study of the fine properties of sets having locally finite distributional fractional perimeter. We refine the characterization of their blow-ups and prove a Leibniz rule for the intersection of sets with locally finite distributional fractional perimeter with sets with finite fractional perimeter. As a byproduct, we provide a description of non-local boundaries associated with the distributional fractional perimeter.

On sets with finite distributional fractional perimeter

TL;DR

The paper advances the theory of sets with locally finite distributional fractional perimeter by sharpening blow-up descriptions, establishing a fractional Leibniz rule for intersections with finite fractional perimeter sets, and clarifying the structure of non-local boundaries. It develops a detailed framework using spaces, fractional gradients , and non-local gradients , proving precise decomposition formulas for under intersections and products, together with tight control via the local fractional perimeter . A refined blow-up analysis shows that tangent sets at fractional reduced boundary points are half-space–type in the limit, with a 1D slice controlling the structure, and introduces the effective fractional reduced boundary , which aligns with classical boundary notions under appropriate regularity. Finally, the work clarifies the support and nature of the singular part of the fractional variation, relating it to and establishing consequences for the geometry of non-local boundaries, including Ball-type explicit computations of the nonlocal gradient on balls.

Abstract

We continue the study of the fine properties of sets having locally finite distributional fractional perimeter. We refine the characterization of their blow-ups and prove a Leibniz rule for the intersection of sets with locally finite distributional fractional perimeter with sets with finite fractional perimeter. As a byproduct, we provide a description of non-local boundaries associated with the distributional fractional perimeter.
Paper Structure (16 sections, 15 theorems, 110 equations)

This paper contains 16 sections, 15 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. The distributional fractional perimeter
  3. Leibniz rules
  4. Analysis of blow-ups
  5. Analysis of non-local boundaries
  6. Proofs of the results
  7. Proof of Theorem 1.1
  8. Proof of Theorem 1.3
  9. Proof of Corollary 1.4
  10. Proof of Theorem 1.6
  11. Proof of Proposition 1.8
  12. Proof of Proposition 1.9
  13. Proof of Theorem 1.7
  14. Proof of Theorem 1.10
  15. Proof of Theorem 1.12
  16. ...and 1 more sections

Key Result

Theorem 1.1

If $\chi_E\in BV^{\alpha,\infty}_{\mathrm{loc}}(\mathbb{R}^n)$ and $P_\alpha(F)<+\infty$, then $\chi_{E\cap F}\in BV^{\alpha,\infty}_{\mathrm{loc}}(\mathbb{R}^n)$, with and Consequently, we have and In addition, if $F$ is also bounded, then $\chi_{E\cap F}\in BV^{\alpha,\infty}(\mathbb{R}^n) \cap L^1(\mathbb{R}^n)$ and

Theorems & Definitions (19)

  • Theorem 1.1: Intersection with $W^{\alpha,1}$ set
  • Definition 1.2: Non-local fractional $\alpha$-gradient measure
  • Theorem 1.3: Conditional Leibniz rule in $BV^{\alpha,\infty}_{\mathrm{loc}}$
  • Corollary 1.4: The measure $D^\alpha_{\mathrm{NL}}(\chi_E,\chi_E)$
  • Theorem 1.5: Existence and rigidity of blow-ups
  • Theorem 1.6: Refined convergence
  • Theorem 1.7: Characterization of blow-ups
  • Proposition 1.8: Null derivative
  • Proposition 1.9: Splitting
  • Theorem 1.10: Support of $D^\alpha_{\mathrm{s}}\chi_E$
  • ...and 9 more