Let $\Omega\subseteq \mathbb{R}^n$ be a domain and let $u$ be a harmonic function on $\Omega$ such that the gradient of $u$ squared integrates to $1$. How can we obtain pointwise upper bounds on $u(x)$ depending only on the distance of the point $x$ from the boundary of $\Omega$?
I obtained this problem from Michael Dafermos' page on Princeton Generals (see here, second question under Analysis). I find this question really interesting and highly appreciate any insight on this.