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Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains

Adriano Prade

TL;DR

This work proves sharp boundary regularity for linear nonlocal elliptic equations with kernels comparable to the fractional Laplacian on Reifenberg flat domains under zero exterior data. The authors develop a barrier-based induction scheme complemented by a compactness argument to upgrade initial Hölder-type regularity to $C^{s-\varepsilon}$ for any $\varepsilon>0$, uniformly in the domain geometry when the flatness parameter is small. Key tools include GMT-type measure estimates near the boundary, a barrier lemma for $\updelta^{\varepsilon}$-type supersolutions, and a scale-wise comparison principle. The results extend boundary Hölder regularity theory to rougher, Reifenberg flat domains, aligning nonlocal regularity with the classical local theory and broadening applicability in analysis and applications.

Abstract

We prove sharp boundary regularity of solutions to nonlocal elliptic equations arising from operators comparable to the fractional Laplacian over Reifenberg flat sets and with null exterior condition. More precisely, if the operator has order $2s$ then the solution is $C^{s-\varepsilon}$ regular for all $\varepsilon>0$ provided the flatness parameter is small enough. The proof relies on an induction argument and its main ingredients are the construction of a suitable barrier and the comparison principle.

Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains

TL;DR

This work proves sharp boundary regularity for linear nonlocal elliptic equations with kernels comparable to the fractional Laplacian on Reifenberg flat domains under zero exterior data. The authors develop a barrier-based induction scheme complemented by a compactness argument to upgrade initial Hölder-type regularity to for any , uniformly in the domain geometry when the flatness parameter is small. Key tools include GMT-type measure estimates near the boundary, a barrier lemma for -type supersolutions, and a scale-wise comparison principle. The results extend boundary Hölder regularity theory to rougher, Reifenberg flat domains, aligning nonlocal regularity with the classical local theory and broadening applicability in analysis and applications.

Abstract

We prove sharp boundary regularity of solutions to nonlocal elliptic equations arising from operators comparable to the fractional Laplacian over Reifenberg flat sets and with null exterior condition. More precisely, if the operator has order then the solution is regular for all provided the flatness parameter is small enough. The proof relies on an induction argument and its main ingredients are the construction of a suitable barrier and the comparison principle.

Paper Structure

This paper contains 4 sections, 9 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Barriers
  4. Boundary regularity

Key Result

Theorem 1.2

Let $L$ be of the type operators-sym and $\varepsilon \in (0,s)$. Let $\Omega \subset \mathbb{R}^n$ be a $(\eta, r_0)$-Reifenberg flat domain for some $r_0>0$, $f \in L^{\infty}(\Omega)$ and $u$ be the weak solution to Then, there exist $\eta_0 = \eta_0(n,s,\lambda, \Lambda, \varepsilon)>0$ and $C= C(n,s, \lambda, \Lambda, \varepsilon)>0$ such that for all $\eta \leq \eta_0$ we have $u \in C^{s-\

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2: Weak solution
  • Definition 2.3: Distributional solution
  • Theorem 2.4: Weak maximum principle
  • Theorem 2.5: Interior regularity
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 12 more