arXiv subjects
Publications and source records attributed to Evgeny Sevost'yanov.
For some class of mappings, which are generalization of space quasiisometries, an upper estimate for a measure of image of a ball is obtained. As consequence, it is obtained one analog of Schwartz lemma for mappings mentioned above. Results of the paper are applicable in Sobolev and Orlicz--Sobolev classes.
A local behavior of closed open discrete mappings of Orlicz--Sobolev classes in ${\Bbb R}^n,$ $n\ge 3,$ is studied. It is proved that, mappings mentioned above have continuous extension to isolated boundary point $x_0$ of a domain $D\setminus\{x_0\}$ whenever $n-1$ degree of its inner dilatation has $FMO$ (finite mean oscillation) at the point and, besides that, limit sets of $f$ at $x_0$ and $\partial D$ are disjoint. Another sufficient condition of possibility of continuous extension is a divergence of some integral
The paper is devoted to the study of mappings with non--bounded characteristics of quasiconformality. We investigate the interconnection between the classes of the so-called ring $Q$-mappings and lower ring $Q$-mappings. It is proved that open discrete lower ring $Q$-mappings are ring $Q^{n-1}$-mappings at fixed point. As consequence we obtain the equicontinuity of the class of the open discrete Orlicz--Sobolev mappings with finite distortion at $n\ge 3.$
For mappings of finite distortion actively investigated last 15--20 years, problems of a so-called lower order are discussed. It is proved that, mappings with finite length distortion $f:D\rightarrow {\Bbb R}^n,$ $n\ge 2,$ which have locally integrable other dilatation in degree $α>n-1,$ and have a finite asymptotic value are of a uniformly lower bounded lower order.
There are investigated problems connected with local and boundary properties of Orlicz--Sobolev classes of finite distortion which are actively studied last time. It is showed that, a locally uniform limit of local homeomorphisms of Orlicz--Sobolev class is a local homeomorphism or a constant whenever it's dilatations satisfy corresponding restrictions. Besides that, at some additional conditions, it is showed that, a locally uniform limit of the mappings mentioned above is a local homeomorphism or a constant.
The present paper is devoted to the study of classes of mappings with non-bounded characteristic of quasiconformality. It is obtained a result on normal families of the open discrete mappings $f:D\rightarrow {\Bbb C}\setminus\{a, b\}$ of the class $W_{loc}^{1, 1}$ having a finite distortion and omitting two fixed values $a\ne b$ in ${\Bbb C},$ maximal dilatations of which has a majorant of the class of finite mean oscillation at every point. In particular, the result mentioned above holds for the so-called $Q$-mappings and is an analog of known Montel theorem for analytic functions.
We consider the generic discrete open mappings in ${\mathbb R}^n$ under which the perturbation of extremal lengths of curve collections is controlled integrally via $\int Q(x)η^p(|x-x_0|) dm(x)$ with $n-1<p<n$, where $Q$ is a measurable function on ${\mathbb R}^n$ and $\int\limits_{r_1}^{r_2} η(r) dr \ge 1$ for any $η$ on a given interval $[r_1,r_2].$ We proved that the family of all open discrete mappings of above type is normal under appropriate restrictions on the majorant $Q.$
The present paper is devoted to the study of classes of mappings with non--bounded characteristics of quasiconformality. It is proved that the normal families of mappings distorting the families of mappings in ${\Bbb R}^n$ by special way, have the logarithmic order of growth in the neighborhood of every point. There are proved some sufficient conditions of normality of such mappings $f:D\rightarrow \bar{{\Bbb R}^n},$ $n\ge 2,$ omitting the values of some set $E_f$ with some constraints of the type $c(E_f)\ge δ,$ $δ\in {\Bbb R},$ where $c(\cdot)$ is the special set function.
The paper is devoted to the study of mappings with non--bounded characteristics of quasiconformality. The analog of the theorem about radius injectivity of locally quasiconformal mappings was proved for some class of mappings. There are found sharp conditions under which the so called local $Q$--homeomorphisms are injective in some neighborhood of a fixed point.
The present paper is devoted to the study of space mappings, which are more general than quasiregular. The so--called modulus inequalities for some class of mappings are obtained. In particular, the analogues of the well--known Poletskii and Väisälä inequalities were proved.
Various theorems on convergence of general space homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring $Q$--homeomorphisms are obtained. In particular, it was established by us that a family of all ring $Q$--homeomorphisms $f$ in ${\Bbb R}^n$ fixing two points is compact provided that the function $Q$ is of finite mean oscillation. These results will have wide applications to Sobolev's mappings.
The present paper is devoted to the study of space mappings, which are more general than quasiregular. The analogue of the known Väisälä inequality for the special types of curves was proved for the open, discrete, differentiable a.e. mappings having $N,$ $N^{-1}$ and $ACP^{-1}$ properties.
The present paper is devoted to the study of space mappings, which are more general than quasiregular mappings. The questions of the behavior of differentiable mappings having the so--called $N,$ $N^{-1},$ $ACP$ and $ACP^{-1}$ -- properties are studied in the work. Under some additional conditions, it is showed that the modulus of such mappings $f$ can be more than each degree of logarithmic function at every neighborhood of the isolated essential singularity of $f.$
First of all, we prove that open mappings in Orlicz-Sobolev classes $W^{1,ϕ}_{\rm loc}$ under the Calderon type condition on $ϕ$ have the total differential a.e. that is a generalization of the well-known theorems of Gehring-Lehto-Menchoff in the plane and of Väisälä in ${\Bbb R}^n$, $n\geqslant3$. Under the same condition on $ϕ$, we show that continuous mappings $f$ in $W^{1,ϕ}_{\rm loc}$, in particular, $f\in W^{1,p}_{\rm loc}$ for $p>n-1$ have the $(N)$-property by Lusin on a.e. hyperplane. Our examples demonstrate that the Calderon type condition is not only sufficient but also necessary for this and, in particular, there exist homeomorphisms in $W^{1,n-1}_{\rm loc}$ which have not the $(N)$-property with respect to the $(n-1)$-dimensional Hausdorff measure on a.e. hyperplane. It is proved on this base that under this condition on $ϕ$ the homeomorphisms $f$ with finite distortion in $W^{1,ϕ}_{\rm loc}$ and, in particular, $f\in W^{1,p}_{\rm loc}$ for $p>n-1$ are the so-called lower $Q$-homeomorphisms where $Q(x)$ is equal to its outer dilatation $K_f(x)$ as well as the so-called ring $Q_*$-homeomorphisms with $Q_*(x)=[K_{f}(x)]^{n-1}$. This makes possible to apply our theory of the local and boundary behavior of the lower and ring $Q$-homeomorphisms to homeomorphisms with finite distortion in the Orlicz-Sobolev classes.