Unique continuation at the boundary for harmonic functions

Abstract: In a work from 1991 Fang-Hua Lin asked the following question. Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Let $u$ be a function harmonic in $\Omega$ and continuous in $\overline \Omega$ which vanishes et $\Sigma \subset \partial\Omega$ and moreover assume that the normal derivative $\partial_\nu u$ vanishes in a subset of $\Sigma$ with positive surface measure. Is it true that then $u$ is identically zero?

Up to now, the answer was known to be positive for $C^1$-Dini domains, by results of Adolfsson-Escauriaza (1997) and Kukavica-Nystrom (1998). In this talk I will explain a recent work where I show that the result also holds for Lipschitz domains with small Lipschitz constant, and thus in particular for general $C^1$ domains.

analysis of PDEs

Audience: researchers in the topic

PDE seminar via Zoom