@article{Dovhopiatyi_Sevost’yanov_2023, title={On compact classes of solutions of Dirichlet problem in simply connected domains}, volume={58}, url={http://www.matstud.org.ua/ojs/index.php/matstud/article/view/350}, DOI={10.30970/ms.58.2.159-173}, abstractNote={<p>The article is devoted to<br>compactness of solutions of the Dirichlet problem for the Beltrami<br>equation in some simply connected domain. In terms of prime ends, we<br>have proved corresponding results for the case when the maximal<br>dilatations of these solutions satisfy certain integral constraints.<br>The first section is devoted to a presentation of well-known<br>definitions that are necessary for the formulation of the main<br>results. In particular, here we have given a definition of a prime<br>end corresponding to N\"{a}kki’s concept. The research tool that was<br>used to establish the main results is the method of moduli for<br>families of paths. In this regard, in the second section we study<br>mappings that satisfy upper bounds for the distortion of the<br>modulus, and in the third section, similar lower bounds. The main<br>results of these two sections include the equicontinuity of the<br>families of mappings indicated above, which is obtained under<br>integral restrictions on those characteristics. The proof of the<br>main theorem is done in the fourth section and is based on the<br>well-known Stoilow factorization theorem. According to this, an open<br>discrete solution of the Dirichlet problem for the Beltrami equation<br>is a composition of some homeomorphism and an analytic function. In<br>turn, the family of these homeomorphisms is equicontinuous<br>(Section~2). At the same time, the equicontinuity of the family of<br>corresponding analytic functions in composition with some<br>(auxiliary) homeomorphisms reduces to using the Schwartz formula, as<br>well as the equicontinuity of the family of corresponding inverse<br>homeomorphisms (Section~3).</p>}, number={2}, journal={Matematychni Studii}, author={Dovhopiatyi, O. and Sevost’yanov, E.}, year={2023}, month={Jan.}, pages={159-173} }