# On compact classes of solutions of Dirichlet problem in simply connected domains

### Abstract

The article is devoted to

compactness of solutions of the Dirichlet problem for the Beltrami

equation in some simply connected domain. In terms of prime ends, we

have proved corresponding results for the case when the maximal

dilatations of these solutions satisfy certain integral constraints.

The first section is devoted to a presentation of well-known

definitions that are necessary for the formulation of the main

results. In particular, here we have given a definition of a prime

end corresponding to N\"{a}kki's concept. The research tool that was

used to establish the main results is the method of moduli for

families of paths. In this regard, in the second section we study

mappings that satisfy upper bounds for the distortion of the

modulus, and in the third section, similar lower bounds. The main

results of these two sections include the equicontinuity of the

families of mappings indicated above, which is obtained under

integral restrictions on those characteristics. The proof of the

main theorem is done in the fourth section and is based on the

well-known Stoilow factorization theorem. According to this, an open

discrete solution of the Dirichlet problem for the Beltrami equation

is a composition of some homeomorphism and an analytic function. In

turn, the family of these homeomorphisms is equicontinuous

(Section~2). At the same time, the equicontinuity of the family of

corresponding analytic functions in composition with some

(auxiliary) homeomorphisms reduces to using the Schwartz formula, as

well as the equicontinuity of the family of corresponding inverse

homeomorphisms (Section~3).

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*Matematychni Studii*,

*58*(2), 159-173. https://doi.org/10.30970/ms.58.2.159-173

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