Sobolev homeomorphism with zero Jacobian almost everywhere

Anisotropic Sobolev homeomorphisms

“…loc -homeomorphism in the plane, 1 ≤ p < 2 whose Jacobian vanishes a.e., has been recently constructed by Hencl [8]; such a mapping satisfies the assumptions of Theorem 1.3. If in Theorem 1.3 we add the hypothesis J f > 0 a.e., we obtain the identities (1.7) and (1.8) using the area formula (see Sections 2 and 3).…”

Orientation-Preserving Young Measures

“…Of course, under regularity assumptions, the positivity of the Jacobian itself relates to (at least local) non-interpenetration of matter; however, for deformations of Sobolev regularity with exponent p below the dimension, the positivity of the Jacobian is not even necessary for injectivity [Hen11] and this question lies outside the scope of the present work. Nevertheless, the natural question of characterizing those Young measures that are generated by sequences of gradients of strictly orientation-preserving maps has so far remained open.…”

Jacobian of weak limits of Sobolev homeomorphisms

“…Indeed, there exists a sequence of Sobolev homeomorphisms f k with J(x, f k ) = a.e., converging weakly in W ,p (Ω, ℝ n ), ⩽ p < n, to the mapping f(x) = x. Let us briefly sketch this using the construction from [10]: we cover Ω by cubes of diameter less than k and on each cube we follow the construction from [10] to obtain a homeomorphism with zero Jacobian a.e. It is possible to make the W ,p -norm of the sequence uniformly bounded and hence find a weakly convergent subsequence.…”