Quasi-linear fractional-order operators in Lipschitz domains

2023/05/28 by Borthagaray, Juan Pablo, Li, Wenbo, Nochetto, Ricardo H. · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2305.17818

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 Ω 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.

Cited by

• Finite Elements with weighted bases for the fractional Laplacian
• Nonlinear nonlocal equations in Reifenberg flat domains

Related

• Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation
• Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains
• Fractional-order operators: Boundary problems, heat equations
• Regularity results for quasilinear elliptic problems driven by the fractional Φ-Laplacian operator
• Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability