Boundary unique continuation on $C^1$-Dini domains and the size of the singular set

TL;DR

This paper establishes bounds on the size of the singular set of harmonic functions in $C^1$-Dini domains, extending boundary unique continuation results by overcoming the lack of monotonicity of frequency functions.

Contribution

It provides the first estimate of the singular set's Minkowski content for harmonic functions in Dini domains without relying on monotonicity of frequency functions.

Findings

Bound on the $(d-2)$-dimensional Minkowski content of the singular set.

Extension of boundary unique continuation results to $C^1$-Dini domains.

Method to handle non-monotonic frequency functions at boundary and interior points.

Abstract

Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in \overline{D}: u(X) = |\nabla u(X)| = 0\} $ and give an estimate of its $(d-2)$-dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of $u$ centered at $0$. Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.