Fractional‐order operators on nonsmooth domains
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Fractional‐order operators on nonsmooth domains. / Abels, Helmut; Grubb, Gerd.
In: Journal of the London Mathematical Society, Vol. 107, No. 4, 2023, p. 1297-1350.Research output: Contribution to journal › Journal article › Research › peer-review
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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