Fractional‐order operators on nonsmooth domains

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Fractional‐order operators on nonsmooth domains. / Abels, Helmut; Grubb, Gerd.

I: Journal of the London Mathematical Society, Bind 107, Nr. 4, 2023, s. 1297-1350.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

Harvard

Abels, H & Grubb, G 2023, 'Fractional‐order operators on nonsmooth domains', Journal of the London Mathematical Society, bind 107, nr. 4, s. 1297-1350. https://doi.org/10.1112/jlms.12712

APA

Abels, H., & Grubb, G. (2023). Fractional‐order operators on nonsmooth domains. Journal of the London Mathematical Society, 107(4), 1297-1350. https://doi.org/10.1112/jlms.12712

Vancouver

Abels H, Grubb G. Fractional‐order operators on nonsmooth domains. Journal of the London Mathematical Society. 2023;107(4):1297-1350. https://doi.org/10.1112/jlms.12712

Author

Abels, Helmut ; Grubb, Gerd. / Fractional‐order operators on nonsmooth domains. I: Journal of the London Mathematical Society. 2023 ; Bind 107, Nr. 4. s. 1297-1350.

Bibtex

@article{bd913a3ef3a1402c90c21fbcba2af923,
title = "Fractional‐order operators on nonsmooth domains",
abstract = "The fractional Laplacian(−Δ)푎,푎∈(0,1),anditsgen-eralizations to variable-coefficient2푎-order pseudodif-ferential operators푃, are studied in퐿푞-Sobolev spacesof Bessel-potential type퐻푠푞. For a bounded open setΩ⊂ℝ푛, consider the homogeneous Dirichlet problem:푃푢 = 푓inΩ,푢=0inℝ푛⧵Ω. We find the regularityof solutions and determine the exact Dirichlet domain퐷푎,푠,푞(the space of solutions푢with푓∈퐻푠푞(Ω))incaseswhereΩhas limited smoothness퐶1+휏,for2푎 < 휏 <∞,0⩽푠<휏−2푎. Earlier, the regularity and Dirichletdomains were determined for smoothΩby the sec-ond author, and the regularity was found in low-orderH{\"o}lder spaces for휏=1by Ros-Oton and Serra. The퐻푠푞-results obtained now when휏<∞arenew,evenfor(−Δ)푎. In detail, the spaces퐷푎,푠,푞are identified as푎-transmission spaces퐻푎(푠+2푎)푞(Ω), exhibiting estimates interms ofdist(푥, 휕Ω)푎near the boundary.The result has required a new development of methodsto handle nonsmooth coordinate changes for pseudod-ifferential operators, which have not been availablebefore; this constitutes another main contribution ofthe paper",
author = "Helmut Abels and Gerd Grubb",
year = "2023",
doi = "10.1112/jlms.12712",
language = "English",
volume = "107",
pages = "1297--1350",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "Oxford University Press",
number = "4",

}

RIS

TY - JOUR

T1 - Fractional‐order operators on nonsmooth domains

AU - Abels, Helmut

AU - Grubb, Gerd

PY - 2023

Y1 - 2023

N2 - The fractional Laplacian(−Δ)푎,푎∈(0,1),anditsgen-eralizations to variable-coefficient2푎-order pseudodif-ferential operators푃, are studied in퐿푞-Sobolev spacesof Bessel-potential type퐻푠푞. For a bounded open setΩ⊂ℝ푛, consider the homogeneous Dirichlet problem:푃푢 = 푓inΩ,푢=0inℝ푛⧵Ω. We find the regularityof solutions and determine the exact Dirichlet domain퐷푎,푠,푞(the space of solutions푢with푓∈퐻푠푞(Ω))incaseswhereΩhas limited smoothness퐶1+휏,for2푎 < 휏 <∞,0⩽푠<휏−2푎. Earlier, the regularity and Dirichletdomains were determined for smoothΩby the sec-ond author, and the regularity was found in low-orderHölder spaces for휏=1by Ros-Oton and Serra. The퐻푠푞-results obtained now when휏<∞arenew,evenfor(−Δ)푎. In detail, the spaces퐷푎,푠,푞are identified as푎-transmission spaces퐻푎(푠+2푎)푞(Ω), exhibiting estimates interms ofdist(푥, 휕Ω)푎near the boundary.The result has required a new development of methodsto handle nonsmooth coordinate changes for pseudod-ifferential operators, which have not been availablebefore; this constitutes another main contribution ofthe paper

AB - The fractional Laplacian(−Δ)푎,푎∈(0,1),anditsgen-eralizations to variable-coefficient2푎-order pseudodif-ferential operators푃, are studied in퐿푞-Sobolev spacesof Bessel-potential type퐻푠푞. For a bounded open setΩ⊂ℝ푛, consider the homogeneous Dirichlet problem:푃푢 = 푓inΩ,푢=0inℝ푛⧵Ω. We find the regularityof solutions and determine the exact Dirichlet domain퐷푎,푠,푞(the space of solutions푢with푓∈퐻푠푞(Ω))incaseswhereΩhas limited smoothness퐶1+휏,for2푎 < 휏 <∞,0⩽푠<휏−2푎. Earlier, the regularity and Dirichletdomains were determined for smoothΩby the sec-ond author, and the regularity was found in low-orderHölder spaces for휏=1by Ros-Oton and Serra. The퐻푠푞-results obtained now when휏<∞arenew,evenfor(−Δ)푎. In detail, the spaces퐷푎,푠,푞are identified as푎-transmission spaces퐻푎(푠+2푎)푞(Ω), exhibiting estimates interms ofdist(푥, 휕Ω)푎near the boundary.The result has required a new development of methodsto handle nonsmooth coordinate changes for pseudod-ifferential operators, which have not been availablebefore; this constitutes another main contribution ofthe paper

U2 - 10.1112/jlms.12712

DO - 10.1112/jlms.12712

M3 - Journal article

VL - 107

SP - 1297

EP - 1350

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 4

ER -

ID: 370480653