Nonlinear Korn inequalities in Rⁿ and immersions in W^2,p, p>n, considered as functions of their metric tensors in W^1,p

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 W^2,p(Ω), p>n, hence with metric tensors in the space W^1,p(Ω); our point of departure is a crucial comparison theorem between solutions in W^1,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 W^2,p(Ω) can be considered as well-defined functions, up to an isometric equivalence relation R(Ω), of their metric tensors in W^1,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 W^1,p(Ω) into the quotient set W^2,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 W_loc ^1,p(Ω), p> n, into the quotient set W_loc ^2,p(Ω)/R(Ω), and that this mapping is continuous when these spaces are equipped with their natural Fréchet topologies.