Abels, Helmut and Grubb, Gerd (2023) Fractional-order operators on nonsmooth domains. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 107 (4). pp. 1297-1350. ISSN 0024-6107, 1469-7750
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The fractional Laplacian (-Delta)a$(-\Delta )<^>a$, a is an element of(0,1)$a\in (0,1)$, and its generalizations to variable-coefficient 2a$2a$-order pseudodifferential operators P$P$, are studied in Lq$L_q$-Sobolev spaces of Bessel-potential type Hqs$H<^>s_q$. For a bounded open set omega subset of Rn$\Omega \subset \mathbb {R}<^>n$, consider the homogeneous Dirichlet problem: Pu=f$Pu =f$ in omega$\Omega$, u=0$u=0$ in Rn set minus omega$ \mathbb {R}<^>n\setminus \Omega$. We find the regularity of solutions and determine the exact Dirichlet domain Da,s,q$D_{a,s,q}$ (the space of solutions u$u$ with f is an element of Hqs(omega over bar )$f\in H_q<^>s(\overline{\Omega })$) in cases where omega$\Omega$ has limited smoothness C1+tau$C<^>{1+\tau }$, for 2a<tau<infinity$2a<\tau <\infty$, 0 <= s<tau-2a$0\leqslant s<\tau -2a$. Earlier, the regularity and Dirichlet domains were determined for smooth omega$\Omega$ by the second author, and the regularity was found in low-order Holder spaces for tau=1$\tau =1$ by Ros-Oton and Serra. The Hqs$H_q<^>s$-results obtained now when tau<infinity$\tau <\infty$ are new, even for (-Delta)a$(-\Delta )<^>a$. In detail, the spaces Da,s,q$D_{a,s,q}$ are identified as a$a$-transmission spaces Hqa(s+2a)(omega over bar )$H_q<^>{a(s+2a)}(\overline{\Omega })$, exhibiting estimates in terms of dist(x, partial differential omega)a$\operatorname{dist}(x,\partial \Omega )<^>a$ 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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | BOUNDARY-VALUE-PROBLEMS; PSEUDODIFFERENTIAL-OPERATORS; DIRICHLET PROBLEM; MU-TRANSMISSION; REGULARITY; EQUATIONS; HEAT |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Helmut Abels |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 22 Feb 2024 08:01 |
| Last Modified: | 22 Feb 2024 08:01 |
| URI: | https://pred.uni-regensburg.de/id/eprint/59024 |
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