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

TL;DR

This paper establishes a global second order Sobolev regularity estimate for solutions to variable coefficient p-Laplacian type equations in bounded convex domains, extending regularity results under minimal smoothness assumptions on coefficients.

Contribution

It provides the first global second order regularity estimate for p-Laplacian type equations with variable coefficients in convex domains with minimal regularity assumptions.

Findings

Proves a global second order Sobolev regularity estimate for solutions.

Extends regularity results to variable coefficient p-Laplacian equations.

Establishes similar results for certain Lipschitz domains with boundary smoothness and smallness conditions.

Abstract

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.