Geodesic
Uniform and tangential harmonic approximation of continuous functions on arbitrary sets
Matematičeskie zametki, Tome 9 (1971) no. 2, pp. 131-142

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Necessary and sufficient conditions which must be imposed on a set $E$ are derived, such that functions continuous on $E\subset G$ can be approximated by functions harmonic in a region $G\subset R^n$. 
@article{MZM_1971_9_2_a3,
     author = {A. A. Shaginyan},
     title = {Uniform and tangential harmonic approximation of continuous functions on arbitrary sets},
     journal = {Matemati\v{c}eskie zametki},
     pages = {131--142},
     publisher = {mathdoc},
     volume = {9},
     number = {2},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_9_2_a3/}
}
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A. A. Shaginyan. Uniform and tangential harmonic approximation of continuous functions on arbitrary sets. Matematičeskie zametki, Tome 9 (1971) no. 2, pp. 131-142. http://geodesic.mathdoc.fr/item/MZM_1971_9_2_a3/