Boundary unique continuation in planar domains by conformal mapping

TL;DR

This paper proves boundary unique continuation properties for harmonic functions in planar domains using conformal mapping techniques, improving understanding of critical points and boundary behavior.

Contribution

It introduces a conformal mapping approach to analyze boundary critical points of harmonic functions, extending previous results and providing new size estimates.

Findings

Harmonic functions with boundary vanishing have finitely many critical points in certain domains.

The conformal mapping technique relates boundary critical sets to interior critical sets of transformed functions.

The results improve recent bounds on critical sets near the boundary in planar domains.

Abstract

Let $\Omega\subset\mathbb R^2$ be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary cannot have the norm of the gradient which vanishes on a subset of positive surface measure (arc length). This result is conjectured to be true in higher dimensions by Lin, in Lipschitz domains. Let now $\Omega\subset\mathbb R^2$ be a $C^1$ domain with Dini mean oscillations. We prove that a nontrivial harmonic function which vanishes continuously on a relatively open subset of the boundary $\partial\Omega\cap B_1$ has a finite number of critical points in $\overline\Omega\cap B_{1/2}$. The latter improves some recent results by Kenig and Zhao. Our technique involves a conformal mapping which moves the boundary where the harmonic function vanishes into an interior nodal line of a new harmonic function, after a further reflection. Then, size estimates of the critical set - up to the boundary - of the original harmonic function can be understood in terms of estimates of the \emph{interior} critical set of the new harmonic function and of the critical set - up to the boundary - of the conformal mapping.