  Let $\Omega\subset R^n$ be a bounded convex domain with $n\ge2$. 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 $1<p<\infty$, we build up a global second order
regularity estimate $$\|D[|Du|^{p-2} Du]\|_{L^2(\Omega)}+\|D[
|\sqrt{A}Du|^{p-2} A Du]\|_{L^2(\Omega)} \le C \|f\|_{L^2(\Omega)} $$ for
inhomogeneous $p$-Laplace type equation \begin{equation}
  -\mathrm{div}\big(\langle A Du,Du\rangle ^{\frac{p-2}2} A Du\big)=f
\quad\rm{in }\ \Omega \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.
