Bianalytic functions of Hölder classes in Jordan domains with nonanalytic boundaries

We consider some boundary behavior effect for bianalytic functions related to the Dirichlet problem solvability. There exist such Jordan domains (even with infinitely smooth but not analytic boundaries) where non-constant bianalytic functions can tend to zero near the boundary only sufficiently slow. More precisely, we prove that for any $\alpha$ and $\beta$ such that $0<\alpha<\beta<1$, there exists a Jordan domain $D=D(\alpha,\beta)$ in which there are nontrivial solutions of the homogeneous Dirichlet problem for the class ${\rm Lip_{\alpha}}(\overline{D})$. At that, every boundary arc is a uniqueness set for functions bianalytic in $D$ and belonging to the class ${\rm Lip_{\beta}}(\ovz{D})$.