Abstract
Let Omega, Omega' subset of R-3 be Lipschitz domains, let f(m) : Omega -> Omega' be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and sup(m) integral(Omega) (|D integral(m)|(2) + 1/J(fm)(2)) < infinity. Let f be a weak limit of f(m) in W-1,W-2.
We show that f is invertible a.e., and more precisely that it satisfies the (INV) condition of Conti and De Lellis, and thus that it has all of the nice properties of mappings in this class. Generalization to higher dimensions and an example showing sharpness of the condition 1/J(f)(2) is an element of L-1 are also given.
Using this example we also show that, unlike the planar case, the class of weak limits and the class of strong limits of W-1,W-2 Sobolev homeomorphisms in R-3 are not the same.