On some properties of generalized quasiisometries with unbounded characteristic
We consider a family of the open discrete mappings $f:\; D \rightarrow \overline{\mathbb{R}^n}$ that distort in a special way the $p$ -modulus of families of curves connecting the components of spherical condenser in a domain $D$ in $\mathbb{R}^n$, $p > n — 1,\;\; p < n$, and omitting...
Збережено в:
| Дата: | 2011 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2011
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2724 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | We consider a family of the open discrete mappings $f:\; D \rightarrow \overline{\mathbb{R}^n}$ that distort in a special way the $p$ -modulus of families of curves connecting the components of spherical condenser in a domain $D$ in $\mathbb{R}^n$, $p > n — 1,\;\; p < n$, and omitting a set of positive $p$-capacity. We establish that this family is normal provided that some function realizing the control of the considered distortion of curve family has a finite mean oscillation at every point or only logarithmic singularities of the order, which is not larger than $n − 1$. We prove that, under these
conditions, an isolated singularity $x_0 \in D$ of the mapping $f : D \ \{x_0\} \rightarrow \overline{\mathbb{R}^n}$ is removable and, moreover,
the extended mapping is open and discrete. As applications we obtain analogs of the known Liouville and Sokhotski – Weierstrass theorems. |
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