Convex domains and unique continuation at the boundary
The background to this paper is the following open question concerning a harmonic function \(u\) on a Lipschitz domain \(D\) in \(\mathbb{R}^d\). If \(u\) vanishes continuously on a relatively open subset \(V\) of \(\partial D\) and the normal derivative of \(u\) vanishes on a subset of \(V\) of positive surface measure, does it follow that \(u \equiv 0\)? From work of \textit{F.-H. Lin} [Commun. Pure Appl. Math. 44, 287-308 (1991; Zbl 0734.58045)] the answer is known to be affirmative when \(D\) is a \(C^{1,1}\) domain. The present paper shows that the answer is also affirmative when \(D\) is convex. This is achieved by establishing a doubling property of the form \(\int_{\Gamma (Q,2r)} u^2 \leq M \int_{\Gamma (Q,r)} u^2\), where \(\Gamma (Q,r) = \{X \in D : |X - Q |< r\}\), for suitable \(Q \in V\) and \(r > 0\). An analogous result for solutions of the heat equation in convex cylinders is also obtained.
scientific article; zbMATH DE number 838774