Failure of $L^r$-Calder\'on-Zygmund estimates for the p-Laplace equation for small $r$

Abstract

Let $p \neq 2$. 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 \[ \begin{cases} {\rm div}(|\nabla u|^{p-2} \nabla u) = {\rm div} (f) \quad& \text{in $\mathbb{B}^2$}\\ u=0 &\text{on $\partial \mathbb{B}^2$} \end{cases} \] but \[ \int_{\mathbb{B}^2} |\nabla u|^r \not \leq \Lambda \int_{\mathbb{B}^2} |f|^{\frac{r}{p-1}}. \] This disproves a conjecture by Iwaniec from 1983. The proof adapts recent convex-integration ideas by Colombo-Tione.