Geodesic
On harmonic approximation in the $C^1$-norm
Sbornik. Mathematics, Tome 71 (1992) no. 1, pp. 183-207

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A criterion is established for the possibility of approximation by harmonic functions and, in particular, by harmonic polynomials in the $C^1$-norm on compact subsets of $\mathbf R^n$. This criterion, which is in terms of harmonic $C^1$-capacity in $\mathbf R^n$, yields a natural analog to the theorem of Vitushkin on rational approximation in terms of analytic capacity. 
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     title = {On harmonic approximation in the $C^1$-norm},
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P. V. Paramonov. On harmonic approximation in the $C^1$-norm. Sbornik. Mathematics, Tome 71 (1992) no. 1, pp. 183-207. http://geodesic.mathdoc.fr/item/SM_1992_71_1_a11/