Nonlinear Korn inequalities in Rn and immersions in W2,p, p>n, considered as functions of their metric tensors in W1,p

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

9 Scopus Citations
View graph of relations


Related Research Unit(s)


Original languageEnglish
Pages (from-to)873-906
Journal / PublicationJournal des Mathematiques Pures et Appliquees
Issue number6
Online published4 Mar 2016
Publication statusPublished - Jun 2016


A nonlinear Korn inequality in Rn provides an upper bound of an appropriate distance between two smooth enough immersions defined over an open subset Ω of Rn in terms of the corresponding distance between their metric tensors.Under the assumption that Ω only satisfies the uniform interior cone property, we first establish such a nonlinear Korn inequality for immersions in the space W2,p(Ω), p>n, hence with metric tensors in the space W1,p(Ω); our point of departure is a crucial comparison theorem between solutions in W1,p(Ω) of Pfaff systems, which is due to the second author.Under the assumptions that Ω is simply-connected and has a Lipschitz-continuous boundary, we then show that such immersions in W2,p(Ω) can be considered as well-defined functions, up to an isometric equivalence relation R(Ω), of their metric tensors in W1,p(Ω) if the Riemann curvature tensors of these tensors vanish in D'(Ω). We also show that the mapping defined in this fashion from the space W1,p(Ω) into the quotient set W2,p(Ω)/R(Ω) is locally Lipschitz-continuous.Under the only assumption that Ω is simply-connected, we finally show that one can define an analogous mapping, this time acting from the space Wloc 1,p(Ω), p> n, into the quotient set Wloc 2,p(Ω)/R(Ω), and that this mapping is continuous when these spaces are equipped with their natural Fréchet topologies.

Research Area(s)

  • Metric tensor, Nonlinear Korn inequalities, Pfaff systems, Riemannian geometry

Citation Format(s)