Unique Continuation on Convex Domains

TL;DR

This paper provides improved quantitative estimates on the critical set of harmonic functions in convex domains, enhancing understanding of boundary behavior and analytic continuation beyond existing results.

Contribution

It introduces novel boundary estimates for harmonic functions in convex domains, surpassing previous methods that combined interior and boundary estimates.

Findings

Enhanced dimensional estimates for the critical set on boundary subsets.

Improved understanding of boundary analytic continuation in convex domains.

New techniques for boundary critical set analysis.

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 \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.