Uniform Approximation by Polynomial Solutions of Second-Order Elliptic Equations, and the Corresponding Dirichlet Problem

Conditions for the uniform approximability of functions by polynomial solutions of second-order elliptic equations with constant complex coefficients on compact sets of special form in $\mathbb R^2$ are studied. The results obtained are of analytic character. Conditions of solvability and uniqueness for the corresponding Dirichlet problem are also studied. It is proved that the polynomial approximability on the boundary of a domain is not generally equivalent to the solvability of the corresponding Dirichlet problem.