A note on the critical set of harmonic functions near the boundary

Abstract

Let $u$ be a harmonic function in a $C^1$ domain $D\subset {\mathbb{R}}^d$, which vanishes on an open subset of the boundary. In this note we study its critical set $\{x\in \overline{D}:\nabla u(x)=0\}$. When $D$ is a $C^{1,\alpha }$ domain for some $\alpha \in (0,1]$, we give an upper bound on the $(d-2)$-dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all $C^1$-Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set $\{x\in \overline{D}:u(x)=0=|\nabla u(x)|\}$ (see [KZ1, KZ2]).