Geodesic
Sufficient conditions of $\alpha$-accessibility of domain in nonsmooth case
Problemy analiza, Tome 2 (2013) no. 1, pp. 3-13.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper continues the study of $\alpha$-accessible domains in $\mathbb{R}^n$ in nonsmooth case. A domain $\Omega\subset \mathbb{R}^n, 0\in\Omega$ is $\alpha$-accessible (see [2], [3]), $\alpha\in [0;1)$; if for every point $p\in\partial\Omega$; there exists a number $r=r(p)>0$ such that the cone $K _{+}(p,\alpha,r)=\{x\in \mathbb{\bar{R}}^{n}(p,r) : (x-p,\frac{p}{||p||})\ge ||x-p||\cos\frac{\alpha\pi}{2}\}$ is included in $\Omega'=\mathbb{R}^n\setminus\Omega$. $\alpha$-accessible domains are starlike domains with respect to the inner point zero and satisfy cone condition which is important for applications,such as the theory of integral representations of functions, imbedding theorems, the questions of the boundary behavior of functions, the solvability of Dirichlet problem. Appropriate conditions of $\alpha$-accessibility of domain, defined by the inequality $F(x)0$ for a continuous function $F$ in $\mathbb{R}^n$ are obtained in [1]. There were obtained and some sufficient conditions of $\alpha$-accessibility. In this article, these sufficient conditions have been significantly strengthened. Here is an example of their use. Conditions, obtained in the theorem, and consequences to it are also sufficient for the starlikeness of the set (the case when $\alpha=0$). In the smooth case the criterion for the $\alpha$-accessibility of domain was obtained in [2]. 
@article{PA_2013_2_1_a0,
     author = {K. F. Amozova},
     title = {Sufficient conditions of $\alpha$-accessibility of domain in nonsmooth case},
     journal = {Problemy analiza},
     pages = {3--13},
     publisher = {mathdoc},
     volume = {2},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PA_2013_2_1_a0/}
}
TY  - JOUR
AU  - K. F. Amozova
TI  - Sufficient conditions of $\alpha$-accessibility of domain in nonsmooth case
JO  - Problemy analiza
PY  - 2013
SP  - 3
EP  - 13
VL  - 2
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2013_2_1_a0/
LA  - ru
ID  - PA_2013_2_1_a0
ER  - 
%0 Journal Article
%A K. F. Amozova
%T Sufficient conditions of $\alpha$-accessibility of domain in nonsmooth case
%J Problemy analiza
%D 2013
%P 3-13
%V 2
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2013_2_1_a0/
%G ru
%F PA_2013_2_1_a0
K. F. Amozova. Sufficient conditions of $\alpha$-accessibility of domain in nonsmooth case. Problemy analiza, Tome 2 (2013) no. 1, pp. 3-13. http://geodesic.mathdoc.fr/item/PA_2013_2_1_a0/

[1] Amozova K. F., Starkov V. V., “Alfa-dostizhimye oblasti, negladkii sluchai”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:3 (2013), 3–8

[2] Liczberski P., Starkov V. V., “Domains in $\mathbb{R}^n$ with conical accessible boundary”, J. Math. Anal. Appl., 408:2 (2013), 547–560 | DOI | MR

[3] Liczberski P., Starkov V. V., “Planar $\alpha$-angularly starlike domains, $\alpha$-angularly starlike functions and their generalizations to multi-dimensional case”, 60 years of analytic functions in Lublin in memory of our professors and friends Jan G. Krzyz, Zdzislaw Lewandowski and Wojciech Szapiel (University of Economics and Innovation in Lublin), Innovatio Press Scientific publishing house, Lublin, 2012, 117–124 | MR

[4] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[5] Dolzhenko E. P., “Granichnye svoistva proizvolnykh funktsii”, Izv. AN SSSR, Ser. Matematika, 31 (1967), 3–14 | DOI | Zbl

[6] Adams R. A., Fournier J., “Cone conditions and properties of Sobolev spaces”, J. Math. Anal. Appl., 61 (1977), 713–734 | DOI | MR | Zbl

[7] Zaremba S., “Sur le principe de Direchlet”, Acta Math., 34 (1911), 293–316 | DOI | MR | Zbl

[8] Dudova A. S., “Usloviya zvezdnosti lebegova mnozhestva differentsiruemoi po napravleniyam funktsii”, Sb. nauch. tr. “Matematika. Mekhanika”, 5 (2003), 30–33, Izd-vo Sarat. un-ta, Saratov

[9] Starkov V. V., “Usloviya zvezdoobraznosti oblastei v $\mathbb{R}^n$”, Tr. PetrGU. Ser. Matematika, 2011, no. 18, 70–82 | MR