H\"{o}lder and Lipschitz continuity in Orlicz-Sobolev classes, distortion and harmonic mappings

TL;DR

This paper investigates the conditions under which injective maps in Orlicz-Sobolev classes exhibit H"{o}lder and Lipschitz continuity, including harmonic and planar mappings, with implications for bi-Lipschitz properties.

Contribution

It establishes new criteria for H"{o}lder and Lipschitz continuity of mappings in Orlicz-Sobolev classes, especially harmonic and planar maps, based on growth conditions and regularity of coefficients.

Findings

H"{o}lder continuity under growth conditions on dilatations

Lipschitz continuity under specific restrictions

Bi-Lipschitz property when Beltrami coefficient is H"{o}lder continuous

Abstract

In this article, we consider the H\"{o}lder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the H\"{o}lder continuity of the indicated class of mappings. In particular, under certain special restrictions, we show that Lipschitz continuity of mappings holds. We also consider H\"{o}lder and Lipschitz continuity of harmonic mappings and in particular of harmonic mappings in Orlicz-Sobolev classes. In addition in planar case, we show in some situations that the map is bi-Lipschitzian if Beltrami coefficient is H\"{o}lder continuous.