Continuity properties of weakly monotone Orlicz-Sobolev functions

TL;DR

This paper investigates the regularity and continuity properties of weakly monotone functions within Orlicz-Sobolev spaces, proposing new conditions for continuity and analyzing behavior outside negligible sets.

Contribution

It introduces a new variant of a continuity condition for weakly monotone Orlicz-Sobolev functions that avoids extra assumptions and extends existing regularity results.

Findings

Proposes a new continuity condition avoiding technical assumptions.

Establishes continuity outside sets of zero Orlicz capacity.

Analyzes behavior outside sets of zero Hausdorff measure.

Abstract

The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to H.Lebesgue. It was introduced, in the setting of Sobolev spaces, by J.Manfredi, and thoroughly investigated in the more general framework of Orlicz-Sobolev spaces by diverse authors, including T.Iwaniec, J.Kauhanen, P.Koskela, J.Maly, J.Onninen, X.Zhong. The present paper complements and augments the available theory of pointwise regularity properties of weakly monotone functions in Orlicz-Sobolev spaces. In particular, a variant is proposed in a customary condition ensuring the continuity of functions from these spaces which avoids a technical additional assumption, and applies to certain situations when the latter is not fulfilled. The continuity outside sets of zero Orlicz capacity, and outside sets of (generalized) zero Hausdorff measure, will are also established when everywhere continuity fails.