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On the Dirichlet problem for degenerate Beltrami equations

Authors :
Ryazanov, Vladimir
Salimov, Ruslan
Srebro, Uri
Yakubov, Eduard
Publication Year :
2012

Abstract

We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ to a Beltrami equation $\bar{\partial}f=\mu \partial f$ in a domain $D\subset\Bbb C$ is the so--called lower $Q-$homeomorphism with $Q(z)=K^T_{\mu}(z, z_0)$ where $K^T_{\mu}(z, z_0)$ is the tangent dilatation of $f$ with respect to an arbitrary point $z_0\in {\bar{D}}$ and develop the theory of the boundary behavior of such solutions. Then, on this basis, we show that, for wide classes of degenerate Beltrami equations $\bar{\partial}f=\mu \partial f$, there exist regular solutions of the Dirichlet problem in arbitrary Jordan domains in $\Bbb C$ and pseudoregular and multi-valued solutions in arbitrary finitely connected domains in $\Bbb C$ bounded by mutually disjoint Jordan curves.<br />Comment: 35 pages. arXiv admin note: substantial text overlap with arXiv:1201.5570

Subjects

Subjects :
Mathematics - Complex Variables

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.1210.5910
Document Type :
Working Paper