Sobolev regularity for the nonlocal $(1, p)$-Laplace equations in the superquadratic case
Dingding Li, Chao Zhang
TL;DR
The article analyzes interior Sobolev regularity of weak solutions to the nonlocal $(1,p)$-Laplacian with dual growth in the superquadratic case $p\ge 2$, revealing a dichotomy in regularity governed by $s_p$. Using a finite difference quotient framework in the fractional setting, together with tail estimates, refined energy inequalities, and a Moser-type iteration, the authors obtain explicit Hölder estimates and precise Sobolev regularity: for $s_p\in\left(0,(p-1)/p\right]$ solutions belong to $W^{\gamma,q}_{\mathrm{loc}}$ for all $q\ge p$ and $\gamma<\frac{s_p p}{p-1}$, while for $s_p\in\left((p-1)/p,1\right)$ the gradient exists with $u\in W^{1,q}_{\mathrm{loc}}$ for all $q\ge p$, culminating in Hölder continuity via embeddings. The work carefully balances the nonlocal $1$-growth structure with the $p$-growth term, navigating difficulties from tail behavior and the lack of smoothness in $Z$, to produce sharp local regularity results and quantitative estimates dependent on tail and seminorms. The findings advance the theory of nonlocal growth problems and provide solid regularity benchmarks for mixed-growth nonlocal equations with potential implications for related variational and PDE models.
Abstract
We investigate the interior Sobolev regularity of weak solutions to the nonlocal $(1, p)$-Laplace equations in the superquadratic case $p\ge 2$. As a product, the explicit Hölder continuity estimates of weak solutions are derived. The proof relies on a detailed analysis of the structural characteristics of $(1, p)$-growth in the nonlocal setting, combined with the finite difference quotient method, tail estimates, refined energy estimates, and a Moser-type iteration scheme.