Local bounds for nonlinear higher-order vector fields for the p-Laplace equation
Felice Iandoli, Giuseppe Spadaro, Domenico Vuono
TL;DR
This work addresses higher-order regularity for solutions to the p-Laplace equation $-\,\Delta_p u=f$ in a domain when $p$ is close to $2$. The authors develop a weighted Calderón–Zygmund framework via a regularization $u_\varepsilon$ and Faà di Bruno expansions to obtain uniform weighted $L^q$ estimates for derivatives of order $m$, proving that $|\nabla u|^{m-2}\nabla u\in W^{m-1,q}_{loc}(\Omega)$ and $|\nabla u|^{m-2}D^2u\in W^{m-2,q}_{loc}(\Omega)$ for all $q\ge2$, under $f$ with Sobolev/Hölder regularity and $f\in S_{loc}(\Omega)$. The results are complemented by a localized Moser iteration that yields uniform $L^\infty$ bounds for weighted high-order derivatives, providing quantitative control even near points with $|\nabla u|=0$. The proofs are completed by passing to the limit as $\varepsilon\to0$ and using density arguments to relax regularity assumptions on $f$, obtaining a near-Caldéron–Zygmund-type theory for the nonlinear operator in the vicinity of the linear case $p=2$. The findings deliver robust local regularity for higher-order stress fields in the p-Laplacian setting and enhance understanding of behavior near critical points of $\nabla u$.
Abstract
We study higher regularity for weak solutions of the $p$-Laplace equation $-Δ_p u = f$ in a domain $Ω\subset \mathbb{R}^n$ for $p$ sufficiently close to 2. For $m \ge 3$, assuming that $f$ satisfies suitable Sobolev and Hölder regularity conditions, we prove that the nonlinear quantity $|\nabla u|^{m-2}\nabla u$ belongs to $W^{m-1,q}_{loc}(Ω)$, and that $|\nabla u|^{m-2} D^2u$ belongs to $W^{m-2,q}_{loc}(Ω)$, for any $q\ge 2$. Furthermore, we obtain uniform $L^\infty$ bounds for the weighted $(m-1)$-th derivatives of $|\nabla u|^{m-2}\nabla u$ and $|\nabla u|^{m-2} D^2u$, providing quantitative control even near critical points of $\nabla u$.