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

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 \ 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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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 $v\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ in any bounded smooth domain $\mathrm{\Omega }$, $\mathrm{\parallel }D[(\mathrm{\mid }Dv{\mathrm{\mid }}^2+\epsilon {)}^{\frac{\gamma }{2}}Dv]{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$ is bounded via $\mathrm{\parallel }(\mathrm{\mid }Dv{\mathrm{\mid }}^2+\epsilon {)}^{\frac{\gamma }{2}}{\mathrm{\Delta }}_{p,\epsilon }^{N}v{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$, where ${\mathrm{\Delta }}_{p,\epsilon }^{N}v$ 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).