Ring homeomorphisms and prime ends

2015/03/30 by Gutlyanskii, Vladimir, Ryazanov, Vladimir, Yakubov, Eduard
#30D40 #35A23 #35J46 #35J67 #35J70 #35J75 #35Q35 #37E30 #Complex Variables (math.CV) #FOS: Mathematics #Primary 30C62 #Secondary 35A16

paper · doi:10.48550/arxiv.1503.08832

Abstract

We show that every homeomorphic W^1,1_{\rm loc} solution f of a Beltrami equation ∂f=μ ∂ f in a domain D⊆\Bbb C is the so--called ring Q-homeomorphism with Q(z)=K^T_μ(z, z₀) where K^T_μ(z, z₀) is the tangent (angular) dilatation quotient of the equation with respect to an arbitrary point z₀∈ D. In this connection, we develop the theory of the boundary behavior of the ring Q-homeomorphisms with respect to prime ends. On this basis, we show that, for wide classes of degenerate Beltrami equations ∂f=μ ∂ f, there exist regular solutions of the Dirichlet problem in arbitrary simply connected domains in \Bbb C and pseudoregular and multivalent solutions in arbitrary finitely connected domains in \Bbb C with boundary datum φ that are continuous with respect to the topology of prime ends.

Related

• The Dirichlet problem and prime ends
• On the Dirichlet problem for degenerate Beltrami equations
• On boundary behavior and Dirichlet problem for Beltrami equations
• On compactness of one class of solutions of the Dirichlet problem
• The theory of prime ends and spatial mappings