Bogdan Klishchuk

• PhD
• Fellow at National Academy of Sciences of Ukraine

We consider the class of ring $Q$-homeomorphisms with respect to $p$-modulus in $\mathbb{R}^{n}$ with $p > n$, and obtain lower bounds for limsups of the distance distortions under such mappings. These estimates can be treated as H\"{o}lder's continuity of the inverses near the origin. The sharpness is illustrated by example

We continue to investigate the regular homeomorphic solutions to nonlinear Beltrami equation introduced in Golberg and Salimov (Complex Var Elliptic Equ 65(1):6–21, 2020). Schwarz Lemma type estimates are obtained involving the length-area method. The lower bounds for the inverses are also established.

We study the asymptotic behavior at infinity of ring Q-homeomorphisms with respect to p-modulus for p>n

УДК 517.54, 517.12 Досліджуються кільцеві Q -гомеоморфізми щодо p -модуля при p > 2 на комплексній площині. Отримано нижню оцінку спотворення трансфінітного діаметра образу круга. Розв'язано задачу про мінімізацію функціонала спотворення трасфінітного діаметра круга на деякому класі кільцевих Q -гомеоморфізмів щодо p -модуля.

УДК 517.5 Досліджується поведінка на нескінченності кільцевих Q -гомеоморфізмів щодо p -модуля при p > n .

The paper considers Q-homeomorphisms with respect to the p-module in the complex plane at p > 2. The behavior of such a mapping class at infinity has been analyzed. The lower power order has been estimated.

We study the behavior at infinity of ring Q-homeomorphisms with respect to p-modulus for p > n.KeywordsRing Q-homeomorphisms p-modulus of a family of curvesQuasiconformal mappingsCondenser p-capacity of a condenser Mathematics Subject Classification (2010) 30C65

We study the behavior at infinity of ring $Q$-homeomorphisms with respect to $p$-modulus for $p>n$.

We consider ring Q-homeomorphisms with respect to the p-modulus in the space ℝⁿ for p>n. A lower bound for the volume of the image of a ball under these mappings is obtained. We solve the extreme problems of minimization of functionals of the volume of the image of a ball and the area of the image of a sphere.

Досліджуються регулярні гомеоморфні розв’язки нелінійного рівняння Бельтрамі на степеневий і логарифмічний порядок росту. Побудовані розв’язки, що показують точність порядку росту в знайдених оцінках.

where n ∈ ℕ, n ≥ 2, γ ∈ ℝ⁺, \( {A}_n={\left\{{a}_k\right\}}_{k=1}^n \) is a system of points such that |ak| = 1, a0 = 0, B0, B∞, \( {\left\{{B}_k\right\}}_{k=1}^n \) is a system of pairwise nonoverlapping domains, \( {a}_k\in {B}_k\subset \overline{\mathrm{\mathbb{C}}} \), \( k=\overline{0,n} \), \( \infty \in {B}_{\infty}\subset \overline{\mathrm{...

We study metric properties of ring Q-homeomorphisms with respect to the p-modulus, p > 2, in the complex plane and establish lower bounds for the areas of disks. An extremal problem concerning minimization of the area functional is also solved.

We consider the class of ring Q-homeomorphisms with respect to p-modulus in ℝⁿ with p > n, and obtain a lower bound for the volume of images of a ball under such mappings. In particular, the following theorem is proved in the paper: Let D be a bounded domain in ℝⁿ, n ≥ 2 and let f: D → ℝⁿ be a ring Q-homeomorphism with respect to p-modulus at a poi...

There were obtained some properties of weakly m-convex sets. Various kinds of the problem of shadow were investigated. There was obtained the lower estimation for the number of balls that are necessary to create a shadow at the point of the sphere in three-dimensional Euclidean space

There were obtained some properties of weakly m-convex sets. Various kinds of the problem of shadow were investigated. There was obtained the lower estimation for the number of balls that are necessary to create a shadow at the point of the sphere in three-dimensional Euclidean space

In the work we consider Q-homeomorphisms w.r.t p-modulus on the complex plane as p > 2. We obtain a lower bound for the area of the image of a circle under such mappings. We solve the extremal problem on minimizing the functional of the area of the image of a circle.

Розглянуто кільцеві Q-гомеоморфізми відносно p-модуля на комплексній площині при p > 2. Для таких класів відоб-ражень встановлено оцінки знизу площі образу круга. Розв’язано екстремальну задачу про мінімізацію функціонала площі образу круга.

In this article we consider Q-homeomorphisms with respect to the p-modulus on the complex plane with p>2. It is obtained a lower area estimate for image of discs under such mappings. We solved the extremal problem about minimization of the area functional of images of discs.

The paper is devoted to the investigation of some properties of multivalued mappings in Euclidean spaces. Fixed-point theorems are proved for multivalued mappings whose restrictions to a certain subset in the closure of a domain satisfy a " coacute angle condition " or a " strict coacute angle condition. " Similar results for the restrictions of mu...

The paper is devoted to the investigation of some properties of multivalued mappings in Euclidean spaces. Fixed-point theorems are proved for multivalued mappings whose restrictions to a certain subset in the closure of a domain satisfy a “coacute angle condition” or a “strict coacute angle condition.” Similar results for the restrictions of multiv...

This paper is devoted to studying of some properties of multivalued mappings in Euclidean space. There were proved theorems on a fixed point for multivalued mappings whose restrictions to some subset in the closure of a domain satisfy "a coacute angle condition" or "a strict coacute angle condition". There also were obtained similar results for res...

We obtained for the case of the space H^n an analogical proposition to the theorem about homeomorphic transformation of the space C^n that is strictly invariant on linear convex compact sets. It was formulated the fundamental theorem of affine geometry in H^n.

The solution of one Zamfiresku's problem was obtained. We discuss the unsolved questions related to the Mizel's problem.

Стаття присвячена вивченню деяких властивостей многозначних вiдображень в евклiдовому просторi. Доведено теореми про нерухому точку для многозначних вiдображень, звуження яких на деяку пiдмножину в замиканнi областi евклiдового простору задовольняють "умовi гострого кута" або "умовi строгого гострого кута". Одержанi результати справедливi i у випад...