Fractional-Order Operators on Nonsmooth Domains

Helmut Abels, Gerd Grubb

Published 2020-04-21Version 1

For the fractional Laplacian $(-\Delta )^a$, $a\in(0,1)$, and its nonsmooth pseudodifferential generalizations $P$, even of order $2a$, we establish regularity properties in $L_p$-Sobolev spaces of Bessel-potential type $H^s_p$ for the solutions $u$ of the homogeneous Dirichlet problem on nonsmooth domains $\Omega \subset \mathbb R^n$: $Pu=f$ on $\Omega $, $\operatorname{supp} u\subset\overline\Omega$. A main result is that for bounded $C^{1+\tau }$-domains and operators $P$ with $C^\tau $-dependence on $x$ (any $\tau \in(0,\infty )$), the solutions with $f$ given in $H^s_p(\Omega)$ ($s\in [0,\tau-2a)$, $p\in(1,\infty)$) have $H^{s+2a}_p$-regularity in the interior of the domain and a singular behavior with a factor $d^a$ close to the boundary, where $d(x)\sim \operatorname{dist}(x,\partial\Omega )$ (more precisely, $u$ belongs to a certain $a$-transmission space $H_p^{a(s+2a)}(\overline\Omega)$). This was previously only known for $\tau =\infty $. 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.