Asymptotic behavior of generalized quasiisometries at a point

Abstract

We consider Q-homeomorphisms with respect to the p-modulus, obtain an estimate for the measure of the image of a ball under these mappings, and study the asymptotic behavior at the origin.

Prime ends in theory of mappings with finite distortion in the plane

“…Note also that by Theorem 8.1 the condition (9.3) can be replaced by each of the conditions (8.3) -(8.7). The example in [14] shows that each of the given conditions are not only sufficient but also necessary for continuous extension of f to the boundary.…”

The Beltrami equations and prime ends

“…Note that condition (8.10) is not only sufficient but also necessary for a continuous extension to the boundary of all direct mappings f with integral restrictions of type (8.9), see, e.g., Theorem 5.1 and Remark 5.1 in [22]. In other words, given Φ z 0 which does not satisfy (8.10), one can find a homeomorphic W 1,1 loc solution of (1.1) with condition (8.9) that is not extended to a homeomorphism of…”