Quasi-linear Fractional-Order Operators in Lipschitz Domains
Abstract.
We prove Besov boundary regularity for solutions of the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients on Lipschitz domains \(\Omega\) of \(\mathbb{R}^d\) . Our estimates are consistent with the boundary behavior of solutions on smooth domains and apply to fractional \(p\) -Laplacians and operators with finite horizon. The proof exploits the underlying variational structure and uses a new and flexible local translation operator. We further apply these regularity estimates to derive novel error estimates for finite element approximations of fractional \(p\) -Laplacians and present several simulations that reveal the boundary behavior of solutions.
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Published In
SIAM Journal on Mathematical Analysis
Pages: 4006 - 4039
DOI: 10.1137/23M1575871
ISSN (online): 1095-7154
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© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 30 May 2023
Accepted: 5 March 2024
Published online: 4 June 2024
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Funding Information
Fondo Clemente Estable: 2022-172393
National Science Foundation (NSF): DMS-1908267
Funding: The first author has been supported in part by Fondo Clemente Estable grant 2022-172393. The third author has been supported in part by NSF grant DMS-1908267.
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