Continuity properties of weakly monotone Orlicz–Sobolev functions

Abstract

Abstract The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to Lebesgue. It was introduced, in the framework of Sobolev spaces, by Manfredi, in connection with the analysis of the regularity of maps of finite distortion appearing in the theory of nonlinear elasticity. Diverse authors, including Iwaniecz, Kauhanen, Koskela, Maly, Onninen, Zhong, thoroughly investigated continuity properties of monotone functions in the more general setting of Orlicz–Sobolev spaces, in view of the analysis of continuity, openness and discreteness properties of maps under minimal integrability assumptions on their distortion. The present paper complements and augments the available Orlicz–Sobolev theory of weakly monotone functions. In particular, a variant is proposed in a customary condition ensuring the continuity of functions from this class, which avoids a technical additional assumption, and applies in certain situations when the latter is not fulfilled. The continuity outside sets of zero Orlicz capacity, and outside sets of (generalized) zero Hausdorff measure are also established when everywhere continuity fails.