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

2025/10/16 by Li, Dingding, Zhang, Chao
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.14346

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 (p-1)/(p), we divide the range of the parameter s_p into two distinct scenarios. Specifically, for any s_p∈ (0, (p-1)/(p)] and q≥ p, we demonstrate that the solutions possess W_{\rm loc}^{γ, q}-regularity for all γ∈ (0, (s_p p)/(p-1)) and the W_{\rm loc}^{1, q}-regularity for any s_p∈ ((p-1)/(p), 1) and q≥ 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.

Citations

• Sobolev regularity for the nonlocal (1, p)-Laplace equations in the superquadratic case
• Gradient continuity for the parabolic (1, p)-Laplace system
• Higher Sobolev regularity on the mixed local and nonlocal p-Laplace equations
• Evolution problems with perturbed 1-Laplacian type operators on random walk spaces
• Gradient regularity for (s,p)-harmonic functions
• Continuity of s-minimal functions
• Higher differentiability for the fractional p-Laplacian
• Gradient continuity for the parabolic (1, p)-Laplace equation under the subcritical case
• Solutions of the fractional 1-Laplacian: existence, asymptotics and flatness results
• Regularity for nonuniformly elliptic equations with p,q-growth and explicit x,u-dependence
• Continuity of the spatial gradient of weak solutions to very singular parabolic equations involving the one-Laplacian
• Strongly anisotropic Anzellotti pairings and their applications to the anisotropic p-Laplacian
• A weak solution to a perturbed one-Laplace system by p-Laplacian is continuously differentiable
• Nonuniformly elliptic Schauder theory
• Local Hölder regularity of minimizers for nonlocal variational problems
• (s,p)-harmonic approximation of functions of least W^s,1-seminorm
• Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case
• Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case

Related

• Sobolev regularity for the nonlocal (1, p)-Laplace equations in the superquadratic case
• Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case
• Regularity for the fractional p-Laplace equation
• Higher Sobolev regularity on the mixed local and nonlocal p-Laplace equations
• Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems