Sharpness of the differentiability almost everywhere and capacitary estimates for Sobolev mappings

Mappings of finite distortion: Size of the branch set

“…The positive statement in Theorem 1.2 follows directly from the results in [25,34]. Moreover, the local integrability assumption of the inner distortion function is sharp for the differentiability almost everywhere even in the class of W 1,n−1 -homeomorphisms of finite distortion, see [17]. It is well-known that this is also the right integrability class for the inner distortion function to guarantee that the zero set of the Jacobian has null measure.…”

“…These inequalities were generalized for mappings of finite distortion in W 1,n loc with locally integrable inner distortion by Koskela and Onninen [26]. The optimal assumptions for these inequalities with lower regularity assumptions were further studied in [11,17,34]. As it was observed in [11], these inequalities would follow for a continuous, discrete and open mapping f of finite distortion with even lower regularity assumptions by following the standard arguments in the theory of quasiregular mappings whenever…”

Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

“…see [28]. This condition on integrability of distortion is sharp, meaning for any δ ∈ (0, 1) and n ≥ 3 there exists a homeomorphism f ∈ W 1,n−1 ((−1, 1) n , R n ) such that K I ∈ L δ ((−1, 1) n ) and f is not classically differentiable on a set of positive measure [14]. The a.e.-differentiability of W 1,n−1 -Sobolev maps also holds for continuous, open, and discrete mappings of finite distortion with nonnegative Jacobian if a particular weighted distortion function is integrable [30].…”