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 $\Omega \subset \mathbb{R}^n$. In particular, these estimates improve dimensional estimates on $\{|\nabla u| =0\}$ both in $V \subset \partial \Omega$ and as it \textit{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.