Bianalytic functions of Hölder classes in Jordan domains with nonanalytic boundaries

Abstract

We consider some boundary behavior effect for bianalytic functions related to the Dirichlet problem solvability. It is proved that there exist such Jordan domains (even with infinitely smooth but not analytic boundaries) where non-constant bianalytic functions may 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 $D=D(\alpha ,\beta )$ possessing the following two properties: (i) there exists a non-constant function of the class ${\mathrm {Lip}}_\alpha ({{\overline{D}}})$ which is bianalytic in D and vanishes identically on the boundary $\partial D$ of D; (ii) every arc containing in $\partial D$ is a uniqueness set for functions bianalytic in D and belonging to the class ${\mathrm {Lip}}_\beta ({{\overline{D}}})$.