A remark on gradients of harmonic functions
It is shown that if \(D\) is a domain of class \(C^{1,s}\) in \(\mathbb{R}^d\), where \(s>0\) and \(d\geq 3\), then there exists a non-zero harmonic function \(u\) which is \(C^1\) up to the boundary of \(D\) such that \(u\) and its gradient vanish on a subset of \(\partial D\) of positive measure. This extends a result of \textit{J. Bourgain} and \textit{T. Wolff} [Colloq. Math. 60/61, 253-260 (1990; Zbl 0731.31006)]. The author remarks that minor modifications of his proof show that the theorem remains true in \(C^1\)-Dini domains, but it is an open question whether the theorem holds true in Lipschitz, or even in \(C^1\) domains.
scientific article; zbMATH DE number 813821