Local behaviour of the second order derivatives of solutions to $p$-Laplace equations

TL;DR

The paper studies local regularity for solutions of the nonlinear PDE $-\,\Delta_p u=f$ by establishing $L^{\infty}$-type bounds on the second derivatives weighted by $|\nabla u|^k$ in the near-$2$ regime $|p-2|\ll 1$. The authors regularize the problem with $\varepsilon$, introduce the weighted Hessian quantity $g_\varepsilon=(\varepsilon+|\nabla u_\varepsilon|^2)^{k/2}|u_{\varepsilon,ij}|$, and develop a second linearization plus Moser iteration framework to derive $L^{2^*q}$ estimates for $g_\varepsilon^q$ and ultimately uniform $L^{\infty}$ bounds. A series of a priori estimates, Calderón–Zygmund-type controls, and density arguments allow passage to the limit $\varepsilon\to 0$ and removal of smoothness assumptions on $f$, yielding $| abla u|^k D^2u$ bounded locally. This extends Calderón–Zygmund regularity theory for the $p$-Laplacian to the critical near-$2$ regime and provides quantitative Hessian control under explicit integrability conditions on the source term.

Abstract

We consider the equation $- Δ_{p} u = f(x)$ in $Ω,$ where $Δ_{p}$ is the $p$-Laplace operator. We provide $L^{\infty}$-type estimates for the second derivatives of solutions when $p$ approaches to $2$.

Theorems & Definitions (16)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.1
• Theorem 2.2: beniMRS
• Remark 2.3
• Lemma 3.1
• proof
• Proposition 3.2