Higher Sobolev regularity on the mixed local and nonlocal p-Laplace equations

2025/01/16 by Yuzhou Fang, Dingding Li, Fang, Yuzhou +3 · 1 citation
Mathematics · #Differential Equations and Boundary Problems #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2501.09487

Abstract

We develop a systematic study of the interior Sobolev regularity of weak solutions to the mixed local and nonlocal p-Laplace equations. To be precise, we show that the weak solution u belongs to W^{2, p}_loc and even W^{2, 2}_{\rm loc} Sobolev spaces in the subquadratic case, while |∇ u|^(p-2)/(2)∇ u is of the class W^{1, 2}_loc in the superquadratic scenario, both of which coincide with that of the classical p-Laplace equations. Moreover, an improved higher fractional differentiability and integrability result u∈ W^{1+β, q}_loc is proved in the full range p∈ (1, ∞) for any q∈ [max\p, 2\, ∞) and β∈(0, \frac 2q). The main analytical tools are the finite difference quotient technique, suitable energy method and tail estimates. As far as we know, our results are new within the context of such mixed problems.

Cited by

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

Related

• Regularity for the fractional p-Laplace equation
• 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
• Higher differentiability for the fractional p-Laplacian
• Sobolev regularity for the perturbed fractional 1-Laplace equations in the subquadratic case