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A remark on gradients of harmonic functions. (English) Zbl 0836.31002

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 J. Bourgain and 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.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions

Citations:

Zbl 0731.31006
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