论文标题
诺伊曼（Neumann
The Neumann problem for the fractional Laplacian: regularity up to the boundary
论文作者
论文摘要
我们研究了分数拉普拉斯的Neumann问题的解决方案边界的规律性。我们证明，如果$ u $是$（ - δ）^s u = f $ in $ω$，$ \ MATHCAL n_s u = 0 $ in $ω^c $中的弱解决方案，那么$ u $ is $ c^α$ is $ c^α$ up tp tp tp tp tp th the Borgues tho $α> 0 $。此外，如果$ s> \ frac12 $，我们就会证明c^{2S-1+α}中的$ u \。为了证明这些结果，我们需要在边界上进行一些对数校正的微妙的Moser迭代。我们的方法使我们能够治疗区域分数拉普拉斯的诺伊曼问题，并建立了相同的边界规律性结果。在我们的结果之前，人们对这些诺伊曼问题的内部规律性得到了充分的了解，但是在边界附近，甚至解决方案的连续性也开放。
We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-Δ)^s u=f$ in $Ω$, $\mathcal N_s u=0$ in $Ω^c$, then $u$ is $C^α$ up tp the boundary for some $α>0$. Moreover, in case $s>\frac12$, we then show that $u\in C^{2s-1+α}(\overlineΩ)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.
