Boundary estimates and a Wiener criterion for the fractional Laplacian

Björn, Jana

doi:10.1090/proc/16647

Boundary estimates and a Wiener criterion for the fractional Laplacian
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by Jana Björn

Proc. Amer. Math. Soc. 152 (2024), 1053-1065

DOI: https://doi.org/10.1090/proc/16647

Published electronically: December 18, 2023

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Abstract:

Using the Caffarelli–Silvestre extension, we show for a general open set $\Omega \subset \mathbf {R}^n$ that a boundary point $x_0$ is regular for the fractional Laplace equation $(-\Delta )^su=0$, $0<s<1$, if and only if $(x_0,0)$ is regular for the extended weighted equation in a subset of $\mathbf {R}^{n+1}$. As a consequence, we characterize regular boundary points for $(-\Delta )^su=0$ 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.

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