Boundary behavior and the Dirichlet problem for Beltrami equations

Kovtonyuk, D.; Petkov, I.; Ryazanov, V.; Salimov, R.

doi:10.1090/S1061-0022-2014-01308-8

Boundary behavior and the Dirichlet problem for Beltrami equations
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by D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov and R. R. Salimov
Translated by: the authors

St. Petersburg Math. J. 25 (2014), 587-603

DOI: https://doi.org/10.1090/S1061-0022-2014-01308-8

Published electronically: June 5, 2014

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Abstract:

It is shown that a homeomorphic solution of the Beltrami equation $\bar {\partial }f=\mu {\partial }f$ in the Sobolev class $W^{1,1}_{\mathrm {loc}}$ is a so-called ring and, simultaneously, lower $Q$-homeomorphism with $Q(z)=K_{\mu }(z)$, where $K_{\mu }(z)$ is the dilatation ratio of this equation. On this basis, the theory of the boundary behavior of such solutions is developed and, under certain conditions on $K_{\mu }(z)$, the existence of regular solutions is established for the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. Also, the existence of pseudoregular as well as many-valued solutions is proved in the case of arbitrary finitely connected domains bounded by mutually disjoint Jordan curves.

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