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<p>The article is devoted to the study of the boundary behavior of the mappings satisfying an upper bound for the distortion of the modulus of the families of paths. We study the case when these mappings are open, discrete, but generally not closed in the domain under consideration. We find conditions under which these mappings have a continuous extension to the boundary in terms of prime ends. In addition, under certain conditions, the corresponding classes are equicontinuous in the closure of the domain, where the closure should also be understood in the sense of prime ends.</p>

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ON THE CARATHEODORY THEOREM FOR NON-CLOSED MAPPINGS

  • 期刊论文

  • 发表日期：

    2026年3月21日

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  • Zarina Kovba,
  • Evgeny Sevost’yanov

摘要

The article is devoted to the study of the boundary behavior of the mappings satisfying an upper bound for the distortion of the modulus of the families of paths. We study the case when these mappings are open, discrete, but generally not closed in the domain under consideration. We find conditions under which these mappings have a continuous extension to the boundary in terms of prime ends. In addition, under certain conditions, the corresponding classes are equicontinuous in the closure of the domain, where the closure should also be understood in the sense of prime ends.