The Dirichlet problem and prime ends

TL;DR

This paper develops a theory for the boundary behavior of solutions to Beltrami equations using prime ends, establishing existence results for Dirichlet problems in various domain types under specific conditions.

Contribution

It introduces a new approach to analyze boundary behavior of Beltrami solutions via prime ends and proves existence of solutions in complex domains with boundary data.

Findings

Existence of regular solutions in simply connected domains.

Existence of pseudoregular and multivalent solutions in finitely connected domains.

Conditions on the complex coefficient μ for solution existence.

Abstract

It is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ${\bar{\partial}}f=\mu\,{\partial}f$ of the Sobolev class $W^{1,1}_{\rm loc}$ with respect to prime ends of domains. On this basis, under certain conditions on the complex coefficient ${\mu}$, it is proved the existence of regular solutions of its Dirichlet problem in arbitrary simply connected domains and pseudoregular as well as multivalent solutions in arbitrary finitely connected domains with continuous boundary data in terms of prime ends.