Uniform and $C^1$-approximability of functions on compact subsets of $\mathbb R^2$ by solutions of second-order elliptic equations

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.