  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.
