Uniform approximation of functionsby 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 ellipticequations with constant complex coefficients on compact sets in $\mathbb{R}^2$ (the particular case of harmonicapproximations is not distinguished).The criterion is stated in terms of the unique (scalar) Harvey–Polking capacity related to the leading coefficient of aLaurent-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 operatorson $\mathbb{R}^2$, the problem considered is technically more difficult than the analogous problemfor $\mathbb{R}^d$, $d>2$.

uniform approximation, Vitushkin's scheme, capacities, homogeneous elliptic equations, Carleson measures