Unique continuation on convex domains

Abstract

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 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.