A nonlinear Calderón-Zygmund $L^2$-theory for the Dirichlet problem involving $-|Du{|}^{\gamma }{\Delta }_p^{N}u=f$

Abstract

We establish a nonlinear Calderón-Zygmund $L^2$-theory to the Dirichlet problem $$-|Du|^γΔ^N_p u=f\in L^2(Ω)\quad {\rm in}\quad Ω; \quad u=0 \ \mbox{on $\partialΩ$} $$ for $n\ge 2$, $p>1$ and a large range of $\gamma >-1$, in particular, for all $p>1$ and all $\gamma >-1$ when $n=2$. Here $\Omega \subset {\mathbb{R}}^n$ is a bounded convex domain, or a bounded Lipschitz domain whose boundary has small weak second fundamental form in the sense of Cianchi-Maz'ya (2018). The proof relies on an extension of an Miranda-Talenti \& Cianchi-Maz'ya type inequality, that is, for any $v\in C_0^{\infty }(\Omega )$ in any bounded smooth domain $\Omega$, $\Vert D[(|Dv{|}^2+\epsilon {)}^{\frac{\gamma }{2}}Dv]{\Vert }_{L^2(\Omega )}$ is bounded via $\Vert (|Dv{|}^2+\epsilon {)}^{\frac{\gamma }{2}}{\Delta }_{p,\epsilon }^{N}v{\Vert }_{L^2(\Omega )}$, where ${\Delta }_{p,\epsilon }^{N}v$ is the $\epsilon$-regularization of normalized $p$-Laplacian. Our results extend the well-known Calderón-Zygmund $L^2$-estimate for the Poisson equation, a nonlinear global second order Sobolev estimate for inhomogeneous $p$-Laplace equation by Cianchi-Maz'ya (2018), and a local $W^{2,2}$-estimate for inhomogeneous normalized $p$-Laplace equation by Attouchi-Ruosteenoja (2018).