Boundary unique continuation in planar domains by conformal mapping

Let Ω⊂ R² be a chord arc domain with small constant. We show that a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary cannot have the norm of the gradient which vanishes on a subset of positive surface measure (arc length). This result was previously known to be true, and conjectured in higher dimensions by Lin, in Lipschitz domains. Let now Ω⊂ R² be a C¹ domain with Dini mean oscillations. We prove that a nontrivial harmonic function which vanishes continuously on a relatively open subset of the boundary ∂Ω∩ B₁ has a finite number of critical points in Ω∩ B1/2. The latter improves some recent results by Kenig and Zhao. Our technique involves a conformal mapping which moves the boundary where the harmonic function vanishes into an interior nodal line of a new harmonic function, after a further reflection. Then, size estimates of the critical set - up to the boundary - of the original harmonic function can be understood in terms of estimates of the interior critical set of the new harmonic function and of the critical set - up to the boundary - of the conformal mapping.