Geodesic
Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations
Sbornik. Mathematics, Tome 190 (1999) no. 2, pp. 285-307

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Several necessary and sufficient conditions for the existence of uniform or $C^1$-approximation of functions on compact subsets of $\mathbb R^2$ by solutions of elliptic systems of the form $c_{11}u_{x_1x_1}+2c_{12}u_{x_1x_2}+c_{22}u_{x_2x_2}=0$ with constant complex coefficients $c_{11}$, $c_{12}$ and $c_{22}$ are obtained. The proofs are based on a refinement of Vitushkin's localization method, in which one constructs localized approximating functions by “gluing together” some special many-valued solutions of the above equations. The resulting conditions of approximation are of a topological and metric nature. 
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     author = {P. V. Paramonov and K. Yu. Fedorovskiy},
     title = {Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations},
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P. V. Paramonov; K. Yu. Fedorovskiy. Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations. Sbornik. Mathematics, Tome 190 (1999) no. 2, pp. 285-307. http://geodesic.mathdoc.fr/item/SM_1999_190_2_a5/