Fine Regularity of Solutions to the Dirichlet Problem Associated with the Regional Fractional Laplacian
Received:November 19, 2019  Revised:March 19, 2020
Key Words: regional fractional Laplacian   Dirichlet problem   H\"{o}lder regularity
Fund Project: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).
 Author Name Affiliation Yanyan LI Department of Mathematics and Physics, Shijiazhuang Tiedao University, Hebei 050043, P. R. China
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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 $p>\frac{N}{2-2\alpha}$ 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.