Том 241, № 1 (2019)

We consider the problem of estimation of the functional \( In\left(\upgamma \right)={r}^{\upgamma}\left(B\mathrm{o},0\right)\prod \limits_{k=1}^nr\left({B}_k,{a}_k\right), \) where r(B_k; a_k) is the inner radius of a domain B_k relative to the point a_k, under the condition \( {a}_0=0,\mid {a}_k\mid =1,k=\overline{1,n},{a}_k\in {B}_k\subset \overline{\mathbb{C}}, \) where the domains B_k ∩ B_p = ∅ B_k ∩ B_p = ∅, k ≠ p, k, p = \( \overline{0,n}, \) and the domains B_k; k = \( \overline{1,n}, \) possess a symmetry relative to a unit circle. In some partial cases, this problem was solved in [2–5]. The present work is devoted to the study of the problem for \( \upgamma \in \left(1,{n}^{\frac{1}{3}}\right]\;\mathrm{and}\;n\ge 14. \)

The orbits of slowly perturbed Hamilton systems and the associated ergodic deformations of Lagrange manifolds are studied. The main results are based on the Mather approach [18, 19] to the construction of the homologies of invariant probabilistic measures, which minimize some Lagrange functionals, and on the elliptic Gromov–Salamon–Zehnder–Floer theory [7, 9, 12, 20, 26] of the construction of invariant manifolds. We have constructed the invariant submanifolds, which are the supports of invariant ergodic measures and have a structure of locally homeomorphic metric spaces. We analyze the problem of construction of efficient criteria of their global homeomorphism, which was posed by Professor A. M. Samoilenko during the study of ergodic deformations of nonlinear Hamilton systems and their adiabatic invariants. It is established that the mapping f : X → Y from a linearly connected Hausdorff space X onto a simply connected (in particular, contractible) space Y is a homeomorphism iff f is local and homeomorphic, and the preimage f⁻¹(y) of every point y ∈ Y is a nonempty compact subset in X.

We study the boundary behavior of the classes of ring mappings on Riemannian manifolds, which are a generalization of quasiconformal mappings by Gehring. In terms of the prime ends of regular domains, the theorems of continuous extension of those classes onto the boundary of a domain are presented.

We study the families of mappings such that the inverse ones satisfy an inequality of the Poletskii type in the given domain. It is proved that those families are equicontinuous at the inner points, if the initial and mapped domains are bounded, and the majorant responsible for a distortion of the modulus is integrable. But if the initial domain is locally connected on its boundary, and if the boundary of the mapped domain is weakly flat, then the corresponding families of mappings are equicontinuous at the inner and boundary points.

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space.