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On Hopf's Lemma for sign-changing supersolutions to fractional Laplacian equations

Azahara DelaTorre, Enea Parini

TL;DR

This work clarifies boundary behavior for sign-changing supersolutions of the fractional Laplacian by linking Hopf's Lemma for $u$ to Hopf's Lemma for its Caffarelli-Silvestre extension $U$, under natural domain and positivity assumptions. It establishes a rigorous equivalence between Hopf's Lemma for the nonlocal problem and the local extension problem, and then refines a known dichotomy by Dipierro–Soave–Valdinoci, showing a slightly weaker growth condition suffices when analyzed via the extension. The results unify forward and reverse implications, connect barrier/maximum-principle techniques with the extension framework, and provide a refined criterion for Hopf's Lemma in nonlocal settings that applies to general domains. Overall, the paper advances understanding of nonlocal boundary behavior and offers practical criteria for verifying Hopf-type results through the extension problem.

Abstract

In this paper we investigate the validity of Hopf's Lemma for a (possibly sign-changing) function $u \in H^s_0(Ω)$ satisfying \[ (-Δ)^s u(x) \geq c(x)u(x) \quad \text{in }Ω,\] where $Ω\subset \mathbb{R}^N$ is an open, bounded domain, $c \in L^\infty(Ω)$, and $(-Δ)^s u$ is the fractional Laplacian of $u$. We show that, under suitable assumptions, the validity of Hopf's Lemma for $u$ at a point $x_0 \in \partial Ω$ is essentially equivalent to the validity of Hopf's Lemma for the Caffarelli-Silvestre extension of $u$ at the point $(x_0,0) \in \mathbb{R}^N \times \mathbb{R}^+$. We also provide a slightly more precise characterization of a dichotomy result stated in a recent paper by Dipierro, Soave and Valdinoci.

On Hopf's Lemma for sign-changing supersolutions to fractional Laplacian equations

TL;DR

This work clarifies boundary behavior for sign-changing supersolutions of the fractional Laplacian by linking Hopf's Lemma for to Hopf's Lemma for its Caffarelli-Silvestre extension , under natural domain and positivity assumptions. It establishes a rigorous equivalence between Hopf's Lemma for the nonlocal problem and the local extension problem, and then refines a known dichotomy by Dipierro–Soave–Valdinoci, showing a slightly weaker growth condition suffices when analyzed via the extension. The results unify forward and reverse implications, connect barrier/maximum-principle techniques with the extension framework, and provide a refined criterion for Hopf's Lemma in nonlocal settings that applies to general domains. Overall, the paper advances understanding of nonlocal boundary behavior and offers practical criteria for verifying Hopf-type results through the extension problem.

Abstract

In this paper we investigate the validity of Hopf's Lemma for a (possibly sign-changing) function satisfying where is an open, bounded domain, , and is the fractional Laplacian of . We show that, under suitable assumptions, the validity of Hopf's Lemma for at a point is essentially equivalent to the validity of Hopf's Lemma for the Caffarelli-Silvestre extension of at the point . We also provide a slightly more precise characterization of a dichotomy result stated in a recent paper by Dipierro, Soave and Valdinoci.
Paper Structure (4 sections, 5 theorems, 78 equations)

This paper contains 4 sections, 5 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Hopf's Lemma
  4. On a result by Dipierro-Soave-Valdinoci

Key Result

Theorem 1

Let $\Omega \subset {\mathbb R}^N$ be a bounded, open domain, let $c \in L^\infty(\Omega)$, and let $u \in H^1_0(\Omega)$ solve Let $x_0 \in \partial \Omega$, and suppose that $\Omega$ satisfies an interior ball condition at $x_0$, that is, there exists a ball $B$ centered at $y_0 \in \Omega$ such that $x_0 \in \partial B$. Suppose that $u$ is continuous at $x_0$, and that there exists an open ne

Theorems & Definitions (12)

  • Theorem : Hopf's Lemma
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Proposition 4.1
  • ...and 2 more