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Boundary behavior and the Dirichlet problem for Beltrami equations. (English. Russian original) Zbl 1302.30050
St. Petersbg. Math. J. 25, No. 4, 587-603 (2014); translation from Algebra Anal. 25, No. 4, 101-124 (2013).
Summary: It is shown that a homeomorphic solution of the Beltrami equation \(\bar{\partial }f=\mu\partial f\) in the Sobolev class \(W^{1,1}_{\operatorname{loc}}\) 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.

MSC:
30E25 Boundary value problems in the complex plane
30C62 Quasiconformal mappings in the complex plane
35J46 First-order elliptic systems
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