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Regularity of sets of finite fractional perimeter and nonlocal minimal surfaces in metric measure spaces

Josh Kline

TL;DR

The paper develops a nonlocal regularity theory for sets of finite fractional perimeter in doubling metric measure spaces by linking the finiteness of the $s$-perimeter to boundary size via upper Minkowski content, and by proving a new necessary condition that $∂^*E$ has zero $s$-Hausdorff measure when the characteristic function lies in the Besov class $B^s_{1,1}$. It extends the Euclidean Visintin-type results to metric spaces using hyperbolic fillings, trace/extension between Besov and Newton-Sobolev spaces, and a fractional Poincaré framework. The paper further studies nonlocal minimizers of the functional $\mathcal{J}_Ω^s$ on bounded domains, proving existence, uniform density, and porosity results, which in turn imply boundary regularity refinements for minimizers. Collectively, these results provide a robust, metric-space-compatible picture of how boundary geometry governs nonlocal perimeter and minimization problems with potential applications to phase transitions and nonlocal geometric analysis.

Abstract

In the setting of a doubling metric measure space, we study regularity of sets with finite $s$-perimeter, that is, sets whose characteristic functions have finite Besov energy, with regularity parameter $0<s<1$ and exponent $p=1$. Following a result of Visintin in $\mathbb{R}^n$, we provide a sufficient condition for finiteness of the $s$-perimeter given in terms of the upper Minkowski codimension of the regularized boundary of the set. We also show that if a set has finite $s$-perimeter, then its measure-theoretic boundary has codimension $s$ Hausdorff measure zero. To the best of our knowledge, this result is new even in the Euclidean setting. By studying certain fat Cantor sets, we provide examples illustrating that the converses of these results do not hold in general. In the doubling metric measure space setting, we then consider minimizers of a nonlocal perimeter functional, extending the definition introduced by Caffarelli, Roquejoffre, and Savin in $\mathbb{R}^n$, and prove existence, uniform density, and porosity results for minimizers.

Regularity of sets of finite fractional perimeter and nonlocal minimal surfaces in metric measure spaces

TL;DR

The paper develops a nonlocal regularity theory for sets of finite fractional perimeter in doubling metric measure spaces by linking the finiteness of the -perimeter to boundary size via upper Minkowski content, and by proving a new necessary condition that has zero -Hausdorff measure when the characteristic function lies in the Besov class . It extends the Euclidean Visintin-type results to metric spaces using hyperbolic fillings, trace/extension between Besov and Newton-Sobolev spaces, and a fractional Poincaré framework. The paper further studies nonlocal minimizers of the functional on bounded domains, proving existence, uniform density, and porosity results, which in turn imply boundary regularity refinements for minimizers. Collectively, these results provide a robust, metric-space-compatible picture of how boundary geometry governs nonlocal perimeter and minimization problems with potential applications to phase transitions and nonlocal geometric analysis.

Abstract

In the setting of a doubling metric measure space, we study regularity of sets with finite -perimeter, that is, sets whose characteristic functions have finite Besov energy, with regularity parameter and exponent . Following a result of Visintin in , we provide a sufficient condition for finiteness of the -perimeter given in terms of the upper Minkowski codimension of the regularized boundary of the set. We also show that if a set has finite -perimeter, then its measure-theoretic boundary has codimension Hausdorff measure zero. To the best of our knowledge, this result is new even in the Euclidean setting. By studying certain fat Cantor sets, we provide examples illustrating that the converses of these results do not hold in general. In the doubling metric measure space setting, we then consider minimizers of a nonlocal perimeter functional, extending the definition introduced by Caffarelli, Roquejoffre, and Savin in , and prove existence, uniform density, and porosity results for minimizers.
Paper Structure (14 sections, 22 theorems, 190 equations, 1 figure)

This paper contains 14 sections, 22 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Doubling and connectedness properties
  4. Measure-theoretic boundary, Minkowski content, and Hausdorff measure
  5. Newton-Sobolev, BV, and Besov classes
  6. Poincaré inequalities and consequences
  7. Hyperbolic fillings
  8. Fractional $s$-perimeter and minimizers of the $\mathcal{J}_\Omega^s$ functional
  9. On the boundary size of sets of finite $s$-perimeter
  10. Examples
  11. Existence of minimizers of $\mathcal{J}_\Omega^s$
  12. Regularity of minimizers of $\mathcal{J}_\Omega^s$
  13. Uniform density estimates
  14. Porosity

Key Result

Theorem 1.1

Let $(Z,d,\nu)$ be a metric measure space satsifying the LLC-1 condition, with $\nu$ a doubling measure. Let $E\subset Z$ be a bounded measurable set, and suppose that there exists $t>0$ such that $\overline{\mathcal{M}}^{-t}(\partial^+E)<\infty$. Then $\chi_E\in B^s_{1,1}(Z)$ for all $0<s<t$.

Figures (1)

  • Figure 1: Example \ref{['ex:Mink LLC']}

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.5
  • Theorem 1.7
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 38 more