Quasi-linear Fractional-Order Operators in Lipschitz Domains (Wenbo Li and collaborators)

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.

Publication:

SIAM Journal on Mathematical AnalysisVol. 56, Iss. 3 (2024)

http://dx.doi.org/10.1137/23M1575871

Author:

Juan Pablo Borthagaray

Instituto de Matematica y Estadistica ``Rafael Laguardia", Facultad de Ingenieria, Universidadde la Republica, Montevideo, Uruguay

Wenbo Li

Institute of Computational Mathematics and Scientific/Engineering Computing of the ChineseAcademy of Sciences, Beijing 100190 China

Email: liwenbo@lsec.cc.ac.cn

Ricardo H. Nochetto

Department of Mathematics and Institute for Physical Science and Technology, University ofMaryland, College Park, MD 20742 USA