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

Abstract

The article is devoted tocompactness of solutions of the Dirichlet problem for the Beltramiequation in some simply connected domain. In terms of prime ends, wehave proved corresponding results for the case when the maximaldilatations of these solutions satisfy certain integral constraints.The first section is devoted to a presentation of well-knowndefinitions that are necessary for the formulation of the mainresults. In particular, here we have given a definition of a primeend corresponding to N\"{a}kki's concept. The research tool that wasused to establish the main results is the method of moduli forfamilies of paths. In this regard, in the second section we studymappings that satisfy upper bounds for the distortion of themodulus, and in the third section, similar lower bounds. The mainresults of these two sections include the equicontinuity of thefamilies of mappings indicated above, which is obtained underintegral restrictions on those characteristics. The proof of themain theorem is done in the fourth section and is based on thewell-known Stoilow factorization theorem. According to this, an opendiscrete solution of the Dirichlet problem for the Beltrami equationis a composition of some homeomorphism and an analytic function. Inturn, the family of these homeomorphisms is equicontinuous(Section~2). At the same time, the equicontinuity of the family ofcorresponding analytic functions in composition with some(auxiliary) homeomorphisms reduces to using the Schwartz formula, aswell as the equicontinuity of the family of corresponding inversehomeomorphisms (Section~3).