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Characterization of sets of finite local and non local perimeter via non local heat equation

Andrea Kubin, Domenico Angelo La Manna

TL;DR

This work analyzes the small-time behavior of the energy $\mathcal{E}_t^s(E)=\int_E\int_{E^c} P^s(x-y,t)\,dy\,dx$ driven by the fractional heat equation. By identifying the correct time-space scaling $g_s(t)$, it derives Γ-convergence results that connect the limit energies to the nonlocal $2s$-perimeter for $s<\tfrac{1}{2}$ and the classical De Giorgi perimeter for $s\ge\tfrac{1}{2}$. The authors establish compactness, compute sharp cube-based estimates, and perform blow-up analyses to prove both lower and upper bounds for the Γ-limits, leading to a complete characterization of sets with finite fractional or local perimeters. As a key application, they deduce the local and nonlocal isoperimetric inequalities via Hardy rearrangement arguments, showing that balls minimize the corresponding perimeters. The results illuminate how nonlocal diffusion governs the geometry of sets and provide a variational underpinning for nonlocal threshold dynamics.

Abstract

In this paper we provide a characterization of sets of finite local and non local perimeter via a $Γ-$convergence result. As an application we give a proof of the isoperimetric inequality, both in the local and in the non local case.

Characterization of sets of finite local and non local perimeter via non local heat equation

TL;DR

This work analyzes the small-time behavior of the energy driven by the fractional heat equation. By identifying the correct time-space scaling , it derives Γ-convergence results that connect the limit energies to the nonlocal -perimeter for and the classical De Giorgi perimeter for . The authors establish compactness, compute sharp cube-based estimates, and perform blow-up analyses to prove both lower and upper bounds for the Γ-limits, leading to a complete characterization of sets with finite fractional or local perimeters. As a key application, they deduce the local and nonlocal isoperimetric inequalities via Hardy rearrangement arguments, showing that balls minimize the corresponding perimeters. The results illuminate how nonlocal diffusion governs the geometry of sets and provide a variational underpinning for nonlocal threshold dynamics.

Abstract

In this paper we provide a characterization of sets of finite local and non local perimeter via a convergence result. As an application we give a proof of the isoperimetric inequality, both in the local and in the non local case.
Paper Structure (14 sections, 19 theorems, 150 equations)

This paper contains 14 sections, 19 theorems, 150 equations.

Table of Contents

  1. Notation
  2. Compactness
  3. Proof of Theorem \ref{['Compthm']} when $s \in (0,\frac{1}{2}) \cup (\frac{1}{2},1)$
  4. Proof of Theorem \ref{['Compthm']} when $s = \frac{1}{2}$
  5. $\Gamma$-convergence for $s\in (0,\frac{1}{2})$
  6. Proof of Theorem \ref{['mainthm1']}(i).
  7. Proof of Theorem \ref{['mainthm1']}(ii).
  8. Characterization of sets of finite $2s$-fractional perimeter.
  9. $\Gamma$-convergence for $s \in [\frac{1}{2},1)$.
  10. Estimates on cubes
  11. Proof of Theorem \ref{['mainthm2']}(i).
  12. Proof of Theorem \ref{['mainthm2']}(ii).
  13. Characterization of sets of finite perimeter.
  14. Application: The local and the non local isoperimetric inequality

Key Result

Theorem 1

Let $n\geq 2$ and $s\in (0,1)$ and define $\mathbb X^s= \rm{BV}(\mathbb{R}^n)$ if $s\geq \frac{1}{2}$ and $\mathbb X^s=H^{s}(\mathbb{R}^n)$ if $s<\frac{1}{2}$. Then

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2.1
  • proof : Compactness for $2s<1$.
  • proof : Compactness for $2s>1$
  • Theorem 2.2: Compactness in $\mathrm{BV}$
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 25 more