We show that a harmonic function which vanishes continuously on an open set of the boundary of a convex domain cannot have a normal derivative which vanishes on a subset of positive surface measure. We also prove a similar result for caloric functions vanishing on the lateral boundary of a convex cylinder.
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Vilhelm Adolfsson, Luis Escauriaza, Carlos E. Kenig, Convex domains and unique continuation at the boundary. Rev. Mat. Iberoam. 11 (1995), no. 3, pp. 513–525DOI 10.4171/RMI/182