Weak continuity of Jacobians of \( W_{loc}^{1,1} \)- homeomorphisms on Carnot groups

Abstract

The limit of a locally uniformly converging sequence of analytic functions is an analytic function. Yu.G. Reshetnyak obtained a natural generalization of that in the theory of mappings with bounded distortion: the limit of every locally uniformly converging sequence of mappings with bounded distortion is a mapping with bounded distortion, and established the weak continuity ofthe Jacobians. 

      In this article, similar problems are studied for a sequence of Sobolev-class homeomorphisms defined on a domain in a two-step Carnot group. We show that if such a sequence converges to some homeomorphism locally uniformly, the sequence of horizontal differentials of its terms is bounded in \( L_{\nu,loc} \), and the Jacobians of the terms of the sequence are nonnegative almost everywhere, then thesequence of Jacobians converges to the Jacobian of the limit homeomorphism weakly in \( L_{1,loc} \); here \( \nu \) is the Hausdorff dimension of the group.