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.
Categories: math.AP