Modulus estimates of semirings with applications to boundary extension problems
Anatoly Golberg, Toshiyuki Sugawa, Matti Vuorinen
TL;DR
The paper develops a higher-dimensional Teichmüller–Grotzsch modulus framework for semirings, extending boundary-extension results to mappings with finite directional dilatations by leveraging modulus distortion bounds. Central tools include semiring modulus estimates, the Main Lemma bounding moduli via angular/directional dilatations, and a boundary-extension theory that yields Lipschitz and weak Hölder regularity at the boundary. The work weakens regularity assumptions on mappings (beyond classical quasiconformality) and provides quantitative criteria for boundary behavior, separation properties, and behavior at infinity, with potential applications to boundary value problems in geometric function theory. Overall, the results offer a robust modulus-based approach to boundary correspondence for Sobolev-type mappings in higher dimensions.
Abstract
In our previous paper [GSV2020], we proved that the complementary components of a ring domain in $\mathbb{R}^n$ with large enough modulus may be separated by an annular ring domain and applied this result to boundary correspondence problems under quasiconformal mappings. In the present paper, we continue this work and investigate boundary extension problems for a larger class of mappings.