Title: Continuity estimates for $n$-harmonic equations

Authors: Tadeusz Iwaniec and Jani Onninen

Issue: Volume 56 (2007), Issue 2, 805-824

Abstract: We investigate the nonhomogeneous $n$-harmonic equation $$\mbox{div}\, |\nabla u|^{n-2}\nabla u =f$$ for $u$ in the Sobolev space $\mathscr{W}^{1,n}(\Omega)$, where $f$ is a given function in the Zygmund class $\mathscr{L}\log^\alpha \mathscr{L}(\Omega)$. In dimension $n=2$ the solutions are continuous whenever $f$ lies in the Hardy space $\mathscr{H}^1(\Omega)$, so in particular, if $f\in \mathscr{L}\log \mathscr{L}(\Omega)$. We show in higher dimensions that within the Zygmund classes the condition $\alpha > n-1$ is both necessary and sufficient for the solutions to be continuous. We also investigate continuity of the map $f \rightarrow \nabla u$, from $\mathscr{L}\log^\alpha \mathscr{L}(\Omega)$ into $\mathscr{L}^n\log^\beta \mathscr{L}(\Omega)$, for \[ -1 < \beta < \frac{n\alpha}{n-1}-1. \] These and other results of the present paper, though anticipated by simple examples, are in fact far from routine. Certainly, they are central in the $p$-harmonic theory.