Unique Continuation on Convex Domains
Sean McCurdy · arXiv (Cornell University) · 2019
In this paper, we obtain estimates on the quantitative strata of the critical set of non-trivial harmonic functions $u$ which vanish continuously on $V \subset \partial Ω$, a relatively open subset of the boundary of a convex domain $Ω\subset \mathbb{R}^n$. In particular, these estimates improve dimensional estimates on $\{| abla u| =0\}$ both in $V \subset \partial Ω$ and as it \textit{approaches} $V \cap \overlineΩ.$ 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.