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 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 Analysis Volume 56, Issue 3 Jun 2024 

https://doi.org/10.1137/23M1575871

Author:

Juan Pablo Borthagaray 

Instituto de Matemática y Estadística “Rafael Laguardia”, Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay.

Email: jpborthagaray@fing.edu.uy

Wenbo Li 

Institute of Computational Mathematics and Scientific/Engineering Computing of the Chinese Academy 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 of Maryland, College Park, MD 20742 USA.

Email: rhn@umd.edu