Prime ends in theory of mappings with finite distortion in the plane. (English) Zbl 1488.30157
Summary: In the present paper, it is studied the boundary behavior of the so-called lower \(Q\)-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function \(Q(z)\) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec, also to solutions of the Beltrami equations, as well as to finitely bi-Lipschitz mappings that a far-reaching extension of the known classes of isometric and quasiisometric mappings.
MSC:
| 30C62 | Quasiconformal mappings in the complex plane |
| 30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |
| 30D40 | Cluster sets, prime ends, boundary behavior |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
| 35A16 | Topological and monotonicity methods applied to PDEs |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35J67 | Boundary values of solutions to elliptic equations and elliptic systems |
| 35J70 | Degenerate elliptic equations |
| 35J75 | Singular elliptic equations |
Keywords:
boundary behavior; prime ends; mappings of finite distortion; lower \(Q\)-homeomorphisms; finitely bi-Lipschitz mappings; isometric and quasiisometric mappings; Beltrami equationsReferences:
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