Fine Regularity of Solutions to the Dirichlet Problem Associated with the Regional Fractional Laplacian 
Received:November 19, 2019 Revised:March 19, 2020 
Key Word:
regional fractional Laplacian Dirichlet problem H\"{o}lder regularity

Fund ProjectL:Supported by the Natural Science Foundation of Hebei Province (Grant No.A2018210018) and the Science and Technology Research Program of Higher Educational in Hebei Province (Grant No.ZD2019047). 

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Abstract: 
In this paper, we study the H\"{o}lder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian $(\Delta)^{\alpha}_{\Omega}$ on a bounded open set $\Omega\subset \mathds{R}$$(N\geq2)$ with $C^{1,1}$ boundary $\partial\Omega$. We prove that when $f\in L^{p}(\Omega)$, and $g\in C(\partial\Omega)$, the following problem $(\Delta)^{\alpha}_{\Omega}u=f$ in $\Omega$, $u=g$ on $\partial\Omega$, admits a unique weak solution $u\in W^{\alpha,2}(\Omega)\cap C(\overline{\Omega})$, where $p>\frac{N}{22\alpha}$ and $\frac{1}{2}<\alpha<1$. To solve this problem, we consider it into two special cases, i.e., $g\equiv0$ on $\partial\Omega$ and $f\equiv0$ in $\Omega$. Finally, taking into account the preceding two cases, the general conclusion is drawn. 
Citation: 
DOI:10.3770/j.issn:20952651.2021.01.008 
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