On boundary values of three-harmonic Poisson integral on the boundary of a unit disk

Abstract

Let $C_0$ be a curve in a disk $D = \{ | z| < 1\}$ tangential to a circle at the point $z = 1$ and let $C_{\theta}$ be the result of rotation of this curve about the origin $z = 0$ by an angle \theta . We construct a bounded function $u(z)$ three-harmonic in $D$ with zero normal derivatives $\cfrac{\partial u}{\partial n}$ and $\cfrac{\partial 2u}{\partial r_2}$ on the boundary such that the limit along $C_{\theta}$ does not exist for all $\theta , 0 \leq \theta \leq 2\pi $.