Approximation by solutions of elliptic equations and systems

We will deal with problems of uniform and $C^m$-approximations by solutions of second order homogeneous elliptic equations with constant complex coefficients and by solutions of systems of such equations on compact sets in the complex plane. We will start with recent results due to M. Mazalov on uniform approximation by solutions of equations in question with singularities located outside compact sets where the approximation is considered. Later on we will concentrate on the problem of uniform approximation by polynomial solutions of our equations. For instance, we plan to discuss the important open conjecture that the classical Walsh–Lebesgue criterion for uniform approximation by harmonic polynomials remains valid in the case of uniform approximation by polynomial solution of general strongly elliptic equation of the type in question. We also plan to touch upon the case of approximation by polynomial solutions on not strongly elliptic equations, when the possibility of approximation is controlled by certain special analytic characteristics of sets where approximation is considered (the concepts of Nevanlinna and $L$-special domains). At the rest of the talk we will briefly discuss some recent results about approximation by solutions of equations in question in $C^m$-norms.