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Sobolev regularity for the perturbed fractional 1-Laplace equations in the subquadratic case

Dingding Li, Chao Zhang

TL;DR

This work analyzes the Sobolev regularity of locally bounded weak solutions to the perturbed fractional $1$-Laplacian with nonhomogeneous growth in the subquadratic range $p\in(1,2)$. By introducing a threshold $s_p$ and applying a nonlocal finite-difference quotient method together with a Moser-type iteration, the authors obtain two distinct regularity outcomes: for $s_p\in(0,(p-1)/p]$ the solution gains differentiability up to any $\sigma< s_p p/(p-1)$ (i.e., $u\in W^{\gamma,q}_{\text{loc}}$ with $\gamma< s_p p/(p-1)$), while for $s_p\in((p-1)/p,1)$ the solution has $W^{1,q}_{\text{loc}}$ regularity for all $q\ge p$. The analysis hinges on sharp energy estimates that relate the nonlocal $p$-growth to the dominant $1$-growth term, a detailed finite-difference framework, and iterative schemes that propagate regularity. The results mirror the superquadratic case and reveal how the $1$-growth structure alters the differentiability threshold, ultimately yielding Hölder and gradient regularity descriptions under precise parametric regimes. These insights advance the nonlocal regularity theory for singular problems with mixed growth and may inform related nonlocal models in physics and geometry.

Abstract

This work investigates the Sobolev regularity of solutions to perturbed fractional 1-Laplace equations. Under the assumption that weak solutions are locally bounded, we establish that the regularity properties are analogous to those observed in the superquadratic case. By introducing the threshold $\frac{p-1}{p}$, we divide the range of the parameter $s_p$ into two distinct scenarios. Specifically, for any $s_p\in \left(0, \frac{p-1}{p}\right]$ and $q\ge p$, we demonstrate that the solutions possess $W_{\rm loc}^{γ, q}$-regularity for all $γ\in \left(0, \frac{s_p p}{p-1}\right)$ and the $W_{\rm loc}^{1, q}$-regularity for any $s_p\in \left(\frac{p-1}{p}, 1\right)$ and $q\ge p$, respectively. Our analysis relies on the nonlocal finite-difference quotient method combined with a Moser-type iteration scheme, which provides a systematic approach to the regularity theory for such nonlocal and singular problems.

Sobolev regularity for the perturbed fractional 1-Laplace equations in the subquadratic case

TL;DR

This work analyzes the Sobolev regularity of locally bounded weak solutions to the perturbed fractional -Laplacian with nonhomogeneous growth in the subquadratic range . By introducing a threshold and applying a nonlocal finite-difference quotient method together with a Moser-type iteration, the authors obtain two distinct regularity outcomes: for the solution gains differentiability up to any (i.e., with ), while for the solution has regularity for all . The analysis hinges on sharp energy estimates that relate the nonlocal -growth to the dominant -growth term, a detailed finite-difference framework, and iterative schemes that propagate regularity. The results mirror the superquadratic case and reveal how the -growth structure alters the differentiability threshold, ultimately yielding Hölder and gradient regularity descriptions under precise parametric regimes. These insights advance the nonlocal regularity theory for singular problems with mixed growth and may inform related nonlocal models in physics and geometry.

Abstract

This work investigates the Sobolev regularity of solutions to perturbed fractional 1-Laplace equations. Under the assumption that weak solutions are locally bounded, we establish that the regularity properties are analogous to those observed in the superquadratic case. By introducing the threshold , we divide the range of the parameter into two distinct scenarios. Specifically, for any and , we demonstrate that the solutions possess -regularity for all and the -regularity for any and , respectively. Our analysis relies on the nonlocal finite-difference quotient method combined with a Moser-type iteration scheme, which provides a systematic approach to the regularity theory for such nonlocal and singular problems.
Paper Structure (10 sections, 23 theorems, 136 equations)

This paper contains 10 sections, 23 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Basic estimates and embedding results
  5. Algebraic inequalities.
  6. Finite difference technique
  7. Integral estimates
  8. Sobolev regularity
  9. The case $s_p\in \left( 0,\frac{p-1}{p}\right]$
  10. The case $s_p\in \left( \frac{p-1}{p},1\right)$

Key Result

Theorem 1.3

Let $p\in (1, 2)$ and $s_p\in \left(0, \frac{p-1}{p}\right]$. Suppose that $u$ is a locally bounded weak solution to problem 1.1. Then, we have for any $q\geq p$ and $s_p\leq \sigma<\frac{s_pp}{p-1}$. Moreover, there exist constants $C$ and $\kappa$ depending on $N, p, q, s_1, s_p, \sigma$ such that for any ball $B_R \equiv B_R\left(x_0\right) \subset \subset \Omega$ with $R\in (0,1)$ and $r\in (

Theorems & Definitions (39)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Corollary 1.8
  • Lemma 2.1
  • Lemma 2.2: Embedding $W^{\gamma,q}\hookrightarrow C^{0,\gamma-\frac{N}{q}}$
  • ...and 29 more