Uniform approximation of functions by solutions of second order homogeneous strongly elliptic equations on compact sets in ${\mathbb{R}}^2$

We obtain a criterion for the uniform approximability of functions by solutions of second-order homogeneous strongly elliptic equations with constant complex coefficients on compact sets in $\mathbb{R}^2$ (the particular case of harmonic approximations is not distinguished).
The criterion is stated in terms of the unique (scalar) Harvey–Polking capacity related to the leading coefficient of a Laurent-type expansion (this capacity is trivial in the well-studied case of non-strongly elliptic equations).
The proof uses an improvement of Vitushkin's scheme, special geometric constructions, and methods of the theory of singular integrals. In view of the inhomogeneity of the fundamental solutions of strongly elliptic operators on $\mathbb{R}^2$, the problem considered is technically more difficult than the analogous problem for $\mathbb{R}^d$, $d>2$.