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

Let u be a harmonic function in a C¹ domain D⊂ Rᵈ, which vanishes on an open subset of the boundary. In this note we study its critical set ∈ D̄: ∇ u(x) = 0 \. When D is a C1,α domain for some α ∈ (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 C¹-Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set ∈ D̄: u(x) = 0 = |∇ u(x)| \ (see [KZ1, KZ2]).