Unique continuation on convex domains

In this paper, we obtain estimates on the quantitative strata of the critical set of non-trivial harmonic functions u which vanish continuously on V C 8 center dot, a rel-atively open subset of the boundary of a convex domain center dot C 18n. In particular, these estimates improve dimensional estimates on {IVuI = 0} both in V C 8 center dot and as it approaches V n center dot. These estimates are not obtainable by naively combining inte-rior and boundary estimates, and represent a significant improvement upon existing results for boundary analytic continuation in the convex case.