let $u$ be a p-harmonic function in $\Omega \subset \mathbb R^N$.
that is $-div(|\nabla u|^{p-2}\nabla u)=-\Delta_p u=0$ in $\Omega$.
We already know that the set $\{u=0\}$ is locally a $C^{1,\alpha}$ hypersurface at the points where $\nabla u\neq 0$.
What can be said about the overall regularity of the set $\partial \{u\neq 0\}$. this will also include the points where gradient of u is zero.