Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant

Abstract

Let (Formula presented.) be a (Formula presented.) 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 (Formula presented.) and continuous in (Formula presented.), which vanishes in a relatively open subset (Formula presented.); moreover, the normal derivative (Formula presented.) vanishes in a subset of (Formula presented.) with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.