Fine Regularity of Solutions to the Dirichlet Problem Associated with the Regional Fractional Laplacian

DOI：10.3770/j.issn:2095-2651.2021.01.008

 作者 单位 李艳艳 石家庄铁道大学数理系, 河北 石家庄 050043

本文主要研究有界开集$\Omega$上,局部分数阶拉普拉斯算子$(-\Delta)^{\alpha}_{\Omega}$的狄利克雷问题的弱解的H\"{o}lder正则性问题,其中$\Omega\subset \mathds{R}~(N\geq2)$, 具有 $C^{1,1}$ 边界,该边界记为$\partial\Omega$.我们证明了,在$f\in L^{p}(\Omega)$和$g\in C(\partial\Omega)$的条件下,下述狄利克雷问题:$(-\Delta)^{\alpha}_{\Omega}u=f$ (在$\Omega$内), $u=g$ （在$\partial\Omega$上), 有唯一弱解 $u\in W^{\alpha,2}(\Omega)\cap C(\overline{\Omega})$, 其中 $p>\frac{N}{2-2\alpha}$ 且 $\frac{1}{2}<\alpha<1$.为解决该问题,我们先考虑两种特殊情况,即:在$\partial\Omega$ 上$g\equiv0$和在$\Omega$内$f\equiv0$,并得出相应结论.最后根据上面两种情况得出一般结论.

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.