Geodesic
Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 120-165

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For a function continuous on a compact set $X\subset\mathbb R^3$ and harmonic inside $X$, we obtain a criterion of uniform approximability by functions harmonic in a neighborhood of $X$ in terms of the classical harmonic capacity. The proof is based on an improved localization scheme of A. G. Vitushkin, on a special geometric construction, and on the methods of the theory of singular integrals. 
@article{TM_2012_279_a9,
     author = {M. Ya. Mazalov},
     title = {Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {120--165},
     publisher = {mathdoc},
     volume = {279},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2012_279_a9/}
}
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M. Ya. Mazalov. Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 279 (2012), pp. 120-165. http://geodesic.mathdoc.fr/item/TM_2012_279_a9/