Mappings of generalized finite distortion and continuity

Abstract

AbstractWe study continuity properties of Sobolev mappings , , that satisfy the following generalized finite distortion inequality for almost every . Here and are measurable functions. Note that when , we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion , where a sharp condition for continuity is that is in the Zygmund space for some . We also show that one can slightly relax the boundedness assumption on to an exponential class with , and still obtain continuous solutions when with . On the other hand, for all with , we construct a discontinuous solution with and , including an example with and .