An example of a nonconstant bianalytic function vanishing everywhere on a nowhere analytic boundary

Abstract

A Jordan domain with a smooth nowhere analytic boundary and a function that is bianalytic in this domain, belongs to the class Lip1 in its closure, vanishes on the boundary, but is not identically zero are constructed.

Free boundary problems in the spirit of Sakai’s theorem

“…A lot of results are already known for conformal maps in model spaces when the inner function θ has a Blaschke part. In fact, the boundaries of such domains may be C 1 but not C 1,α for any α ∈ (0, 1) [2, Theorem 2], may be unrectifiable [11, Example 1] or nowhere analytic [10], or may be rectifiable but admitting no such functions continuable up to the boundary [5,Example 5.8]. Furthermore, the Hausdorff dimension of Nevanlinna domains' boundaries can be any number from 1 up to 2 [3, Theorem 1].…”

О Задаче Дирихле Для Полианалитических Функций

“…Заметим, что ∂D не может пересекаться ни с какой аналитической дугой на множестве положительной длины (в противном случае непостоянная бианалитическая функция z − A(z) равна нулю на этом множестве). Вместе с тем, существуют области с нигде не аналитической границей, в которых есть непостоянные бианалитические функции, равные нулю на границе [4]. В § § 2-4 исследуется вопрос о разрешимости краевой задачи Дирихле для полианалитических функций, поставленный, в частности, в [5; § 4].…”

Nevanlinna domains with large boundaries

“…Constructing Nevanlinna domains with irregular (for instance nonanalytic, non-smooth, and even more irregular) boundaries is a rather difficult and delicate problem. It was considered in [23,17,2,26,27]. The detailed account of these results will be given in Section 2 below.…”

“…The first example of N -domain with nowhere analytic (but rather smooth) boundary was constructed in [23]. Later on, several constructions of N -domains with boundaries belonging to the class C 1 , but not to the class C 1,α , α ∈ (0, 1), were obtained in [17] and [2].…”