Boundary estimates and a Wiener criterion for the fractional Laplacian
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914518346498048 |
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| author | Björn, Jana |
| author_facet | Björn, Jana |
| contents | Using the Caffarelli--Silvestre extension, we show for a general open set $\Om\subset\R^n$ that a boundary point $x_0$ is regular for the fractional Laplace equation $(-Δ)^su=0$, $0<s<1$, if and only if $(x_0,0)$ is regular for the extended weighted equation in a subset of $\R^{n+1}$. As a consequence, we characterize regular boundary points for $(-Δ)^su=0$ by a Wiener criterion involving a Besov capacity. A decay estimate for the solutions near regular boundary points and the Kellogg property are also obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_04364 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Boundary estimates and a Wiener criterion for the fractional Laplacian Björn, Jana Analysis of PDEs Mathematics Subject Classification (2020): Primary: 35R11, Secondary: 35B65, 35J25 Using the Caffarelli--Silvestre extension, we show for a general open set $\Om\subset\R^n$ that a boundary point $x_0$ is regular for the fractional Laplace equation $(-Δ)^su=0$, $0<s<1$, if and only if $(x_0,0)$ is regular for the extended weighted equation in a subset of $\R^{n+1}$. As a consequence, we characterize regular boundary points for $(-Δ)^su=0$ by a Wiener criterion involving a Besov capacity. A decay estimate for the solutions near regular boundary points and the Kellogg property are also obtained. |
| title | Boundary estimates and a Wiener criterion for the fractional Laplacian |
| topic | Analysis of PDEs Mathematics Subject Classification (2020): Primary: 35R11, Secondary: 35B65, 35J25 |
| url | https://arxiv.org/abs/2107.04364 |