Mappings of finite distortion: Compactness.

Abstract

We study mappings f: › ! R n whose distortion functions Kl(x;f) , l = 1; 2;:::;ni 1 , are in general unbounded but subexponentially integrable. The main result is the weak compactness principle. It asserts that a family of mappings with prescribed volume integral R › J(x;f)dx , and with given subexponential norm k l p KlkExpA of a distortion function, is closed under weak convergence. The novelty of this result is twofold. Firstly, it requires integral bounds on the distortions Kl(x;f) which are weaker than those for the usual outer distortion. Secondly, the category of subexponential bounds is optimal to fully describe the compactness principle for mappings of unbounded distortion, even when outer distortion is used.