On Hopf's Lemma for sign-changing supersolutions to fractional Laplacian equations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915858075353088 |
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| author | DelaTorre, Azahara Parini, Enea |
| author_facet | DelaTorre, Azahara Parini, Enea |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19843 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Hopf's Lemma for sign-changing supersolutions to fractional Laplacian equations DelaTorre, Azahara Parini, Enea Analysis of PDEs 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. |
| title | On Hopf's Lemma for sign-changing supersolutions to fractional Laplacian equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.19843 |