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

Xavier Tolsa

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arXiv · 2004.10721

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

Xavier Tolsa

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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 $0$.

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Xavier Tolsa. 2020-04-22. Unique continuation at the boundary for harmonic functions in $C^1$ domains and Lipschitz domains with small constant. https://arxiv.org/abs/2004.10721

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