Lectures on Mappings of Finite Distortion

Sobolev homeomorphisms in W^1,k and the Lusin's condition (N) on k-dimensional subspaces

“…On two remaining coordinates the mapping is the identity. The idea of the construction goes back to Ponomarev [14] (see also [9,Section 5] and [7,Chapter 4.3]). …”

“…[1]) it is moreover natural to expect that the mapping f : (0, 1) n → R n is moreover a homeomorphism because of the interpenetration of the matter and the same injectivity assumption is needed in Geometric Function Theory (see [7,8] and references given there). For these homeomorphisms it is crucial to know that sets of zero H n measure are mapped to sets of zero H n measure as this corresponds to the fact that no material is "created" during our deformation.…”

“…For these homeomorphisms it is crucial to know that sets of zero H n measure are mapped to sets of zero H n measure as this corresponds to the fact that no material is "created" during our deformation. It was shown that this n-dimensional Lusin (N ) condition is true for homeomorphisms in W 1,n ((0, 1) n , R n ) by Reshetnyak [15] (see also [11,14] or [7,Theorem 4.10] for sharpness).…”

“…[6], [7,Section A.8] and [12]) which are extremely important tools in the area. Each Sobolev mapping is approximatively differentiable a.e.…”

“…It is known by the Cesari's construction that there is a mapping g ∈ W 1,2 ((0, 1) 2 , R 2 ) which fails the two dimensional Lusin (N ) condition (see [2] or [7,Theorem 4.3]) but unfortunately this mapping is far from being injective. We correct this by pushing the next steps of this construction to higher level in the third dimension and we construct a homeomorphismg : (0, 1) 2 → R 3 which fails the 2-dimensional Lusin (N ) condition.…”