Approximation by solutions of elliptic equations and systems: classical problems & current view

First we plan to consider the problems on approximation of functions by solutions of second-order elliptic equations with constant coefficients and systems of such equations, where the approximation is considered in spaces of continuous and $C^m$-smooth functions, $m>0$, on compact sets in $\mathbb R^N$, $N\geqslant2$. The roots of these problems traced to the classical works by Runge, Walsh, Mergelyan, Keldysh, Vitushkin concerning uniform approximations by harmonic and holomorphic functions. Nowadays in the general context of approximation by solutions of elliptic equations and systems there are several topical open questions. We plan to present a brief survey of this themes and discuss some of the open questions mentioned above.
Next we are going to present the results of recent joint work with P. Paramonov (Lomonosov Moscow State University) where we study capacities related with second-order elliptic PDEs with constant complex coefficients and defined in classes of bounded and continuous functions (in terms of these capacities the approximation criteria are stated which are recently obtained in the problem under consideration). Our main aim is to study the question on comparability of these capacities with the classical harmonic ones.