Higher Order Calderon-Zygmund Estimates for the p-Laplace Equation

Abstract

The paper is concerned with higher order Calderon-Zygmund estimates for the $p$-Laplace equation $$-\text{div}(A(\nabla u)):=-\text{div}(|\nabla u{|}^{p-2}\nabla u)=-\text{div}F,1<p<\infty .$$ We are able to transfer local interior Besov and Triebel-Lizorkin regularity up to first order derivatives from the force term $F$ to the flux $A(\nabla u)$. For $p\ge 2$ we show that $F\in B_{\rho ,q}^s$ implies $A(\nabla u)\in B_{\rho ,q}^s$ for any $s\in (0,1)$ and all reasonable $\rho ,q\in (0,\infty ]$ in the planar case. The result fails for $p<2$. In case of higher dimensions and systems we have a smallness restriction on $s$. The quasi-Banach case $0<\min \{\rho ,q\}<1$ is included, since it has important applications in the adaptive finite element analysis. As an intermediate step we prove new linear decay estimates for $p$-harmonic functions in the plane for the full range $1<p<\infty$.