Nonlinear commutators for the fractional p-Laplacian and applications

TL;DR

This paper establishes a nonlinear commutator estimate for the fractional p-Laplacian, demonstrating higher differentiability of solutions and classical regularity of weak solutions, with applications to the convergence of fractional harmonic maps.

Contribution

It introduces a novel nonlocal commutator estimate for the fractional p-Laplacian, leading to new regularity results and convergence properties for fractional harmonic maps.

Findings

Solutions to certain degenerate nonlocal equations are higher differentiable.

Weak fractional p-harmonic functions are actually classical.

Sequences of bounded fractional harmonic maps converge strongly outside finitely many points.

Abstract

We prove a nonlocal, nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions. For the fractional $p$-Laplace operator it implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weak fractional $p$-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded $\frac{n}{s}$-harmonic maps converge strongly outside at most finitely many points.