Let Ω ⊂ R n , n ≥ 4, be a domain and 1 ≤ p < [n/2], where [a] stands for the integer part of a. We construct a homeomorphism f ∈ W 1,p ((−1, 1) n , R n ) such that J f = det Df > 0 on a set of positive measure and J f < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f k such that f k → f in W 1,p .

Let Ω ⊂ R n be a domain, n ≥ 2. We show that a continuous, open and discrete mapping f ∈ W 1,n−1 loc (Ω, R n ) with integrable inner distortion is differentiable almost everywhere on Ω. As a corollary we get that the branch set of such a mapping has measure zero.

Abstract. Let X be an open set in R n and suppose that f : X → R n is a Sobolev homeomorphism. We study the regularity of f −1 under the L p -integrability assumption on the distortion function K f . First, if X is the unit ball and p > n−1, then the optimal local modulus of continuity of f −1 is attained by a radially symmetric mapping. We show that this is not the case when p n − 1 and n 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for |Df −1 | in terms of the L p -integrability assumptions of K f .

We show that every L-BLD-mapping in a domain of R n is a local homeomorphism if L < √ 2 or KI (f ) < 2. These bounds are sharp as shown by a winding map.