On Dirichlet problem for second-order elliptic equations in the plane and uniform approximation problems for solutions of such equations
We consider the Dirichlet problem for solutions to general second-order homogeneous elliptic equations with constant complex coefficients. We prove that any Jordan domain with $C^{1,\alpha}$-smooth boundary, $0<\alpha<1$, is not regular with respect to the Dirichlet problem for any not strongly elliptic equation $\mathcal Lf=0$ of this kind, which means that for any such domain $G$ it always exists a continuous function on the boundary of $G$ that can not be continuously extended to the domain under consideration to a function satisfying the equation $\mathcal Lf=0$ therein. Since there exists a Jordan domain with Lipschitz boundary that is regular with respect to the Dirichlet problem for bianalytic functions, this result is near to be sharp. We also consider several connections between Dirichlet problem for elliptic equations under consideration and problems on uniform approximation by polynomial solutions of such equations.