Unique continuation at the boundary for harmonic functions in $C^1$ domains and Lipschitz domains with small constant

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $\Omega$ and continuous in $\overline{\Omega }$ which vanishes in a relatively open subset $\Sigma \subset \partial \Omega$ and moreover the normal derivative ${\partial }_{\nu }u$ vanishes in a subset of $\Sigma$ with positive surface measure, then $u$ is identically $0$.

More detailed explanation in some argument involving integration by parts and in Remark 3.3. An additional appendix with a self-contained proof of Lemma 4.3, whose proof was not included in the paper previously

References in corpus (2)

• Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps
• Boundary unique continuation on $C^1$-Dini domains and the size of the singular set

Cited by in corpus (2)

• Boundary unique continuation on $C^1$-Dini domains and the size of the singular set
• Łojasiewicz Inequalities and Generic Smoothness of Nodal Sets of Solutions to Elliptic PDE