Аннотация:
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 of
the 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
$L_{\nu,\mathrm{loc}}$, and the Jacobians of the terms of the sequence are nonnegative almost everywhere, then the
sequence of Jacobians converges to the Jacobian of the limit homeomorphism weakly in $L_{\nu,\mathrm{loc}}$; here $\nu$
is the Hausdorff dimension of the group.
Ключевые слова и фразы:
Carnot group, Sobolev mapping, Jacobian, continuity property.
Полный текст:
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