arXiv 2007.048082020-07math.AP

区域分数拉普拉斯算子：边界正则性

Regional fractional Laplacians: Boundary regularity

Mouhamed Moustapha Fall

备注：: Minor changes has been done. To appear in the "Journal of Differential Equations"

摘要：

利用$Ω\subset \mathbb{R}^N$研究了一类涉及所谓区域分数Lapacians $(-Δ)^s_Ω$的方程解的边界正则性。回想一下，区域分数拉普拉斯算子是由不允许跳出$Ω$的对称稳定过程生成的。我们考虑方程$(-Δ)^s_Ωw(x)=p.v.\int_Ω\frac{w(x)-w(y)}{|x-y|^{N+2s}}\, dy=f(x)$的弱解，对于$s\in (0,1)$，受零Neumann或Dirichlet边界条件的约束。边界条件的定义考虑$w$以及测试功能的分数Sobolev空间$H^s(Ω)$或$H^s_0(Ω)$分别。虽然这些问题的内部正则性很好地理解，但对边界正则性知之甚少，主要是对诺依曼问题。在最优正则性假设下，证明了零Neumann边界条件下的$f\in L^p(Ω)$.作为$2s-N/p>1$、$w\in C^{1,2s-\frac{N}{p}-1}(\overlineΩ)$的结果。对于Dirichlet问题，我们得到了${w}/{δ^{2s-1}}\in C^{1-N/p}(\overlineΩ)$，给出了$p>N$和$s\in (1/2,1)$，其中$δ(x)=\textrm{dist}(x,\partialΩ)$ .为了证明这些结果，我们首先分类所有的解决方案有一定的增长在无穷远时$Ω$是一个半空间和右手边是零。然后，我们继续进行一个良好的爆破和一些紧性参数，以获得结果。

英文摘要：

We study boundary regularity for solutions to a class of equations involving the so called regional fractional Lapacians $(-Δ)^s_Ω$, with $Ω\subset \mathbb{R}^N$. Recall that the regional fractional Laplacians are generated by symmetric stable processes which are not allowed to jump outside $Ω$. We consider weak solutions to the equation $(-Δ)^s_Ωw(x)=p.v.\int_Ω\frac{w(x)-w(y)}{|x-y|^{N+2s}}\, dy=f(x)$, for $s\in (0,1)$, subject to zero Neumann or Dirichlet boundary conditions. The boundary conditions are defined by considering $w$ as well as the test functions in the fractional Sobolev spaces $H^s(Ω)$ or $H^s_0(Ω)$ respectively. While the interior regularity is well understood for these problems, little is known in the boundary regularity, mainly for the Neumann problem. Under optimal regularity assumptions on $Ω$ and provided $f\in L^p(Ω)$, we show that $w\in C^{2s-N/p}(\overline Ω)$ in the case of zero Neumann boundary conditions. As a consequence for $2s-N/p>1$, $w\in C^{1,2s-\frac{N}{p}-1}(\overlineΩ)$. As what concerned the Dirichlet problem, we obtain ${w}/{δ^{2s-1}}\in C^{1-N/p}(\overlineΩ)$, provided $p>N$ and $s\in (1/2,1)$, where $δ(x)=\textrm{dist}(x,\partialΩ)$. To prove these results, we first classify all solutions having a certain growth at infinity when $Ω$ is a half-space and the right hand side is zero. We then carry over a fine blow up and some compactness arguments to get the results.

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摘要：

英文摘要：

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