On Dirichlet problem and uniform approximation by solutions of second-order elliptic systems in R2

We consider the Dirichlet problem for solutions of second-order elliptic systems in the plane that correspond to homogeneous elliptic equations with constant complex coefficients. We prove that any Jordan domain G subset of C with C-1 ,C-alpha-smooth boundary, 0 < alpha < 1, is not regular with respect to the Dirichlet problem for any non-strongly elliptic equation under consideration, which means that there always exists a continuous complex valued function on partial differential G that can not be continuously extended to G to a function satisfying the corresponding equation therein. Since there exists a Jordan domain with Lipschitz boundary, which is regular with respect to the Dirichlet problem for bianalytic functions, the result obtained is near to be sharp. We also consider the problem on uniform approximation of functions by polynomial solutions of homogeneous second-order elliptic equations with constant complex coefficients and give a new proof of the approximation criterion in this problem on Caratheodory compact sets.(c) 2023 Elsevier Inc. All rights reserved.