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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 orderregularity estimate $$\|D[|Du|^{p-2} Du]\|_{L^2(\Omega)}+\|D[|\sqrt{A}Du|^{p-2} A Du]\|_{L^2(\Omega)} \le C \|f\|_{L^2(\Omega)} $$ forinhomogeneous $p$-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 Lipschitzdomain whose boundary is weakly second order differentiable and satisfies somesmallness assumptions.

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