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Plane domains with special cone condition
A. N. Anikiev Petrozavodsk State University,
Lenin Avenue, 33, 185910 Petrozavodsk, Russia.
Abstract:
The paper considers the domains with cone condition in
$\mathbb{C}$.
We say that domain G satisfies the (weak)
cone condition, if
$p+V(e(p),H)\subset{G}$ for all
$p\in{G}$, where
$V(e(p),H)$ denotes
right-angled circular cone with vertex at the origin, a fixed
solution
$\varepsilon$ and a height
$H$,
$0<{H}\leq\infty$, and
depending on the
$p$ vector
$e(p)$ axis direction.
Domains satisfying cone condition play an important role in various
branches of mathematic (e. g. [1], [2], [3] (p. 1076), [4]).
In the paper of P. Liczberski and V. V. Starkov,
$\alpha$–accessible domains were considered,
$\alpha\in[0,1)$, —
the domains, accessible at every boundary point by the cone with
symmetry axis on
$\{pt:t>1\}$.
Unlike the paper of P. Liczberski and V. V. Starkov, here
we consider domains, accessible outside by the cone, which symmetry
axis inclined on fixed angle
$\phi$ to the
$\{pt: t>1\}$,
$0<\|\phi\|<\pi/2$.
In this paper we give criteria for this class of domains when the
boundaries of domains are smooth, and also give a sufficient
condition when boundary is arbitrary.
This article is the full variant of [5], published without proofs.
Keywords:
$(\alpha,\beta)$–accessible domain, cone condition.
MSC: 26A21 Received: 07.07.2014
Language: English
DOI:
10.15393/j3.art.2014.2609
Fulltext:
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