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

Abstract

Let $\Omega \subset R^n$ 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 $1<p<\infty$, we build up a global second order regularity estimate $$\Vert D[|Du{|}^{p-2}Du]{\Vert }_{L^2(\Omega )}+\Vert D[|\sqrt{A}Du{|}^{p-2}ADu]{\Vert }_{L^2(\Omega )}\le C\Vert f{\Vert }_{L^2(\Omega )}$$ 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 }\ 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.