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 $V \subset \partial \Omega$, a relatively open subset of the boundary of a convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$. In particular, these estimates improve dimensional estimates on $\{\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }=0\}$ both in $V \subset \partial \Omega$andasit\mathit{\text{approaches}}$V \cap \overline{\Omega}.$ These estimates are not obtainable by naively combining interior and boundary estimates and represent a significant improvement upon existing results for boundary analytic continuation in the convex case.