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 $\mathcal {C}(u): = \{x \in \overline {D}: \nabla u(x) = 0 \}$. 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 $\mathcal {S}(u): = \{x \in \overline {D}: u(x) = 0 = |\nabla u(x)| \}$ (see [Arch. Ration. Mech. Anal. 245 (2022), pp. 1–88] and [Adv. Nonlinear Stud. 23 (2023)]).