Boundary unique continuation in planar domains by conformal mapping

Abstract Let be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: 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 is conjectured to be true in higher dimensions by Lin, in Lipschitz domains. Let now be a domain with Dini mean oscillations. We prove that a nontrivial harmonic function which vanishes continuously on a relatively open subset of the boundary has a finite number of critical points in . 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.