Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$.
Question: What is the Hausdorff dimension of the critical set $$ \Sigma(u)=\{x\in \Omega\mid \nabla u(x)=0\}? $$ For example, can the dimension be bounded above by $n-2$?
(Note that in general the optimal regularity of $u$ is $C^{1,\alpha}_{\mathrm{loc}}(\Omega)$, and if $p$ satisfies the Cordes condition $1<p<3+\frac{2}{n-2}$, then $u$ is also $W^{2,2}_{\mathrm{loc}}(\Omega)$. See Lindqvist's notes.)
Some results I have found: (In a review article by Magnanini)
- When $n=2$ and $1<p<\infty$ (the planar case), it is known that $\Sigma(u)$ consists of isolated points. So this case is solved. The proofs rely on complex function methods, which I don't see generalizing to higher dimensions.
- When $n\geq 3$ and $p=2$ (the linear case), it is known that $\dim_H \Sigma(u)\leq n-2$. The proofs I've seen for this case rely on the Bers approximation (locally approximating a harmonic function by harmonic polynomials), which I am not sure how to generalize to other values of $p$.
References would be appreciated.
Edit 1: As pointed out in the comments, $\Sigma(u)$ may have dimension at least $n-2$, by taking a $p$-harmonic function on the plane $\mathbb{R}^2$ and extending it trivially to $\mathbb{R}^n$. The question has been modified accordingly.
