Higher differentiability for the fractional $p$-Laplacian

TL;DR

This paper investigates the higher differentiability of solutions to the fractional p-Laplace equation, extending existing results in the superquadratic case and providing new insights in the subquadratic regime, applicable to systems as well.

Contribution

It advances the understanding of regularity for fractional p-Laplace equations, especially in the subquadratic case where previous results were lacking.

Findings

Extended higher differentiability results in the superquadratic case.

Established new regularity results in the subquadratic regime.

Results are consistent with classical p-Laplace equations in the local limit.

Abstract

In this work, we study the higher differentiability of solutions to the inhomogeneous fractional $p$-Laplace equation under different regularity assumptions on the data. In the superquadratic case, we extend and sharpen several previous results, while in the subquadratic regime our results constitute completely novel developments even in the homogeneous case. In particular, in the local limit our results are consistent with well-known higher differentiability results for the standard inhomogeneous $p$-Laplace equation. All of our main results remain valid in the vectorial context of fractional $p$-Laplace systems.