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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