李艳艳.局部分数阶拉普拉斯算子的狄利克雷问题弱解的细正则性[J].数学研究及应用,2021,41(1):69~86 |
局部分数阶拉普拉斯算子的狄利克雷问题弱解的细正则性 |
Fine Regularity of Solutions to the Dirichlet Problem Associated with the Regional Fractional Laplacian |
投稿时间:2019-11-19 修订日期:2020-03-19 |
DOI:10.3770/j.issn:2095-2651.2021.01.008 |
中文关键词: 局部分数阶拉普拉斯算子 狄利克雷问题 H\"{o}lder正则性 |
英文关键词:regional fractional Laplacian Dirichlet problem H\"{o}lder regularity |
基金项目:河北省自然科学基金(Grant No.A2018210018), 河北省高等学校科学技术研究项目(Grant No.ZD2019047). |
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中文摘要: |
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. |
英文摘要: |
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. |
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