TY - JOUR

T1 - WEAK CONTINUITY OF JACOBIANS OF Wν1-HOMEOMORPHISMS ON CARNOT GROUPS

AU - Pavlov, S. V.

AU - Vodop’yanov, S. K.

PY - 2024

Y1 - 2024

N2 - 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 Lν,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 L1,loc; here ν is the Hausdorff dimension of the group.

AB - 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 Lν,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 L1,loc; here ν is the Hausdorff dimension of the group.

KW - Carnot group

KW - Jacobian

KW - Sobolev mapping

KW - continuity property

UR - https://www.mendeley.com/catalogue/04569063-4b62-3159-bce9-65b2cbac94d0/

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85216945860&origin=inward&txGid=b58caee8c066389ba5df9a5565730ce2

U2 - 10.32523/2077-9879-2024-15-4-82-95

DO - 10.32523/2077-9879-2024-15-4-82-95

M3 - Article

VL - 15

SP - 82

EP - 95

JO - Eurasian Mathematical Journal

JF - Eurasian Mathematical Journal

SN - 2077-9879

IS - 4

ER -