A nonlinear Calder\'on-Zygmund $ L^2$-theory for the Dirichlet problem involving $ -|Du|^{\gamma}\Delta^N_p u=f$

Abstract

We establish a nonlinear Calder\'on-Zygmund $L^2$-theory to the Dirichlet problem $$-|Du|^{\gamma}\Delta^N_p u=f\in L^2(\Omega)\quad {\rm in}\quad \Omega; \quad u=0 \ \mbox{on $\partial\Omega$} $$ for $n\ge2$, $ 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 $v\in C^\infty_0(\Omega)$ in any bounded smooth domain $\Omega$, $\|D[(|Dv|^2+\epsilon)^{\frac\gamma 2}Dv]\|_{L^2(\Omega)}$ is bounded via $\|(|Dv|^2+\epsilon)^{\frac\gamma 2} \Delta^N_{p,\epsilon}v \|_{L^2(\Omega)}$, where $\Delta^N_{p,\epsilon}v$ is the $\epsilon$-regularization of normalized $p$-Laplacian. Our results extend the well-known Calder\'on-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).