论文标题
非平滑域的分数运算符
Fractional-Order Operators on Nonsmooth Domains
论文作者
论文摘要
分数laplacian $（ - δ）^a $，$ a \ in（0,1）$及其对可变的$ 2A $ 2A $ -2A $ - 级pseudodifferential operator $ p $ P $的概括，以$ l_q $ -sobolev的空间为bessel-potitential type $ h^s_q $。对于有界的开放集$ω\ subset \ mathbb r^n $，请考虑均质的dirichlet问题：$ pu = f $ in $ω$，$ u = 0 $ in $ \ mathbb r^n \setMinusΩ$。我们找到解决方案的规律性，并确定确切的dirichlet域$ d_ {a，s，q} $（解决方案的空间$ u $ a $ a $ f \ in h_q^s（\overlineΩ$）在$ω$有限的平滑度$ c^{1+τ} $的情况下，对于$ 2a <f fty $ c <fty $ 0.早些时候，第二作者确定了针对平滑$ω$的规律性和差异域，并且Ros-oton和Serra在低阶Hölder空间中以$τ= 1 $的价格发现了规律性。当$τ<\ infty $是新的，即使对于$（ - δ）^a $时，现在获得的$ H_Q^S $ -RESULTS现在获得了。详细说明，空间$ d_ {a，s，q} $被标识为$ a $ a-transmission spaces $ h_q^{a（s+2a）}（\overlineΩ）$，以$ \ permatatorname {dist}（x，\partialΩ）^a $在边界附近显示估计值。结果需要进行新的开发方法来处理以前尚未可用的假差异操作员的非平滑坐标更改；这构成了本文的另一个主要贡献。
The fractional Laplacian $(-Δ)^a$, $a\in(0,1)$, and its generalizations to variable-coefficient $2a$-order pseudodifferential operators $P$, are studied in $L_q$-Sobolev spaces of Bessel-potential type $H^s_q$. For a bounded open set $Ω\subset \mathbb R^n$, consider the homogeneous Dirichlet problem: $Pu =f$ in $Ω$, $u=0$ in $ \mathbb R^n\setminusΩ$. We find the regularity of solutions and determine the exact Dirichlet domain $D_{a,s,q}$ (the space of solutions $u$ with $f\in H_q^s(\overlineΩ)$) in cases where $Ω$ has limited smoothness $C^{1+τ}$, for $2a<τ<\infty $, $0\le s<τ-2a$. Earlier, the regularity and Dirichlet domains were determined for smooth $Ω$ by the second author, and the regularity was found in low-order Hölder spaces for $τ=1$ by Ros-Oton and Serra. The $H_q^s$-results obtained now when $τ<\infty $ are new, even for $(-Δ)^a$. In detail, the spaces $D_{a,s,q}$ are identified as $a$-transmission spaces $H_q^{a(s+2a)}(\overlineΩ)$, exhibiting estimates in terms of $\operatorname{dist}(x,\partialΩ)^a$ near the boundary. The result has required a new development of methods to handle nonsmooth coordinate changes for pseudodifferential operators, which have not been available before; this constitutes another main contribution of the paper.
