Uniform approximation and fine potential theory (Q1804000)

The following theorem is proved: Let \(F\) be a closed subset of \(\mathbb{R}^ N\), \(N\geq 2\). Then \(u\) belongs to the closure in the topology of uniform convergence on \(F\) of all harmonic functions on neighbourhoods of \(F\) if and only if \(u\) is continuous on \(F\) and \(u\) is finely harmonic on the fine interior on \(F\). The case of \(F\) being compact has been proved by \textit{A. Debiard} and \textit{B. Gaveau} see [J. Funct. Anal. 16, 289-304 (1974; Zbl 0297.31004)].