Failure of Calderón-Zygmund Estimates for the p-Laplace Equation

Abstract

Let \(p \ne 2\). For any small enough \(r> \max \{p-1,1\}\) and for any \(\Lambda > 1\) there exists a Lipschitz function u and a bounded vectorfield f such that

$$ {\left\{ \begin{array}{ll} \textrm{div}(|\nabla u|^{p-2} \nabla u) = \textrm{div} (f) \quad & \text {in }\mathbb {B}^2\\ u=0 & \text {on }\partial \mathbb {B}^2 \end{array}\right. } $$

but

$$ \int _{\mathbb {B}^2} |\nabla u|^r \not \le \Lambda \int _{\mathbb {B}^2} |f|^{\frac{r}{p-1}}. $$

This disproves a conjecture by Iwaniec from 1983.