On boundary properties of spatial nonhomeomorphic mappings with an $s$-averaged characteristic

A. N. Malyutina ; K. A. Alipova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 3 (2016), pp. 16-30

Résumé

In this paper, we continue to develop the geometric method of modules of curve families for studying analytical and geometrical properties of nonhomeomorphic mappings with $s$-averaged characteristic. We consider the question of the erasure of special sets under mappings with $s$-averaged characteristic. In this work, in contrast to previous results which require that the mapping is homeomorphic or the capacity of singular points is zero, nonhomeomorphic mappings with $s$-averaged characteristic are considered and a weaker condition is taken as constraints. We generalize the theorem which is known in the case $n = 2$ as Iversen–Tsuji's theorem for the case $n\geqslant3$. There are well-known examples demonstrating the existence of essential singularities for which Hausdorff's measure $\Lambda_\beta\ne0$ at some $\beta\ne0$ for mappings with an $s$-averaged characteristic. The work presents some examples which illustrate distinctive properties of the considered class of mappings. A theorem about the module distortion for families of curves under mappings with allowance for multiplicity and, as a consequence, the characteristic property of the spherical module of families of curves asymptotic to a special boundary set is proved. The mappings are extended to continuous ones if the dimension of the set of singular points $I$ $\mathrm{dim}\, I \leqslant n-2$ and $s > 1$. The results are applicable to many classes of mappings of subclasses $W_n^1(U)$.

Détail

Keywords: spatial mappings with s-averaged characteristics, method of modules, desingularization, estimates of the distortion, asymptotic lifts.