The Beltrami equations and prime ends

Abstract

Abstract.We first study the boundary behavior of ring Q-homeomorphisms in terms of Carathéodory's prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation ∂f = µ ∂f in arbitrary bounded finitely connected domains D of the complex plane C.

On the global behavior of inverse mappings in terms of prime ends

“…Remark 1.8. The possibility of continuous extension of a homeomorphism h : D ′ → D to the mapping h : D ′ → D P in Theorem 1.7 may be established similarly to [GRY,Theorem 6.1]; see also [SalSev,Theorem 2]. Since the proof of this result almost literally repeats the reasoning related to the mentioned publications, we will not give this proof in the present text.…”

On boundary extension of one class of mappings in terms of prime ends

“…Remark 1.1. The possibility of continuous extension of a homeomorphism h : D ′ → D to the mapping h : D ′ → D P in Theorem 1.1 may be established similarly to [GRY,Theorem 6.1]; see also [SalSev,Theorem 2]. Since the proof of this result almost literally repeats the reasoning related to the mentioned publications, we will not give this proof in the present text.…”

О Равностепенной Непрерывности Семейств Отображений С Одним Условием Нормировки

“…Введение. Настоящая статья посвящена изучению отображений с ограниченным и конечным искажением (см., например, [1] и [2]). В частности, мы продолжаем и немного дополняем исследования, проведенные в [3].…”