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$C^{1+α}$ regularity for fractional $p$-harmonic functions

Davide Giovagnoli, David Jesus, Luis Silvestre

TL;DR

This work establishes interior $C^{1,α}$ regularity for solutions to the fractional $p$-Laplacian $(-\Delta_p)^s u=0$ in the regime $p\in[2,2/(1-s))$. The authors develop a three-stage strategy: (i) a De Giorgi-type improvement of flatness for localized directional derivatives via the linearized kernel, (ii) an Ishii–Lions argument to propagate near-constancy of the gradient, and (iii) a nondegenerate regime where classical nonlocal regularity results yield Hölder continuity of the gradient. The analysis hinges on careful control of tail terms, scaling, and a localized linearization $\mathcal{L}_u$, with quantitative estimates that depend only on the dimension, order, and nonlocal parameters. The result resolves a long-standing open problem by proving $C^{1,α}$ regularity for the stated parameter range and provides a robust framework combining nonlocal De Giorgi iterations with variational and viscosity techniques. Overall, the work significantly advances the understanding of nonlocal $p$-Laplacian regularity and opens avenues for further sharp regularity results near critical parameter regimes.

Abstract

We establish interior $C^{1,α}$ regularity estimates for some $α> 0$, for solutions of the fractional $p$-Laplace equation $(-Δ_p)^s u = 0$ when $p$ is in the range $p \in [2,2/(1-s))$.

$C^{1+α}$ regularity for fractional $p$-harmonic functions

TL;DR

This work establishes interior regularity for solutions to the fractional -Laplacian in the regime . The authors develop a three-stage strategy: (i) a De Giorgi-type improvement of flatness for localized directional derivatives via the linearized kernel, (ii) an Ishii–Lions argument to propagate near-constancy of the gradient, and (iii) a nondegenerate regime where classical nonlocal regularity results yield Hölder continuity of the gradient. The analysis hinges on careful control of tail terms, scaling, and a localized linearization , with quantitative estimates that depend only on the dimension, order, and nonlocal parameters. The result resolves a long-standing open problem by proving regularity for the stated parameter range and provides a robust framework combining nonlocal De Giorgi iterations with variational and viscosity techniques. Overall, the work significantly advances the understanding of nonlocal -Laplacian regularity and opens avenues for further sharp regularity results near critical parameter regimes.

Abstract

We establish interior regularity estimates for some , for solutions of the fractional -Laplace equation when is in the range .

Paper Structure

This paper contains 34 sections, 59 theorems, 272 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Previous results
  3. Strategy of the proof
  4. Preliminaries
  5. Viscosity and weak solutions
  6. Tail functions
  7. Scaling
  8. The linearized equation satisfied by directional derivatives
  9. The linearized operator
  10. Localizing the directional derivatives
  11. Reducing the norm of the gradient
  12. Estimating J-Tail
  13. Estimating Jgap
  14. Estimating Jphi
  15. Estimating Jr
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $u \in L_{sp}^{p-1}(\mathbb R^d)$, be a solution to the equation eq:fracplap in the ball $B_2$. Assume that $s \in (0,1)$ and $2 \leq p <2/(1-s)$. Then $u$ is $C^{1,\alpha}$ regular in $B_1$ for some $\alpha > 0$ and the following estimate holds Here $\alpha$ and $C$ are positive constants depending only on $d$, $s$ and $p$. The quantity ${\mathop{\mathrm{Tail}}\limits}_{p-1,sp}(u,2)$ account

Figures (2)

  • Figure 1: Description of the method.
  • Figure 2: Setting of the proof of Lemma \ref{['l:iteration']}.

Theorems & Definitions (104)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 94 more