The Dirichlet problem and prime ends

2015/03/14 by Denis Kovtonyuk, Kovtonyuk, Denis, Igor Petkov +3
Computer Science · Mathematics · #30D40 #35A23 #35J67 #35J70 #35J75 #37E30. Secondary: 35A16 #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Primary: 30C62

paper · pdf · doi:10.48550/arxiv.1503.04306

openalex publication_date 2015/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ∂f=μ ∂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 μ, 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.

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