A global second order Sobolev regularity for $p$-Laplacian type equations with variable coefficients in bounded domains

Let $\mathrm{\Omega }\subset R^n$ be a bounded convex domain with $n\ge 2$. Suppose that $A$ is uniformly elliptic and belongs to $W^{1,n}$ when $n\ge 3$ or $W^{1,q}$ for some $q>2$ when $n=2$. For $1<p<\mathrm{\infty }$, we build up a global second order regularity estimate $$\mathrm{\parallel }D[\mathrm{\mid }Du{\mathrm{\mid }}^{p-2}Du]{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}+\mathrm{\parallel }D[\mathrm{\mid }\sqrt{A}Du{\mathrm{\mid }}^{p-2}ADu]{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}$$ for inhomogeneous $p$-Laplace type equation \begin{equation} -$\mathrm{d}\mathrm{i}\mathrm{v}$\big(\langle A Du,Du\rangle ^{\frac{p-2}2} A Du\big)=f \quad\rm{in }\ Ω\mbox{ with Dirichlet/Neumann $0$-boundary.} \end{equation} Similar result was also built up for certain bounded Lipschitz domain whose boundary is weakly second order differentiable and satisfies some smallness assumptions.