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 \x D: u (x) = 0 \. When D is a C^1, domain for some (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 \x D: u (x) = 0 = | u (x) | \ (see KZ1, KZ2).