Daniel Campbell - Pathological Sobolev homeomorphisms in GFT and NE

Sobolev homeomorphisms are the natural choice for minimization problems in non-linear elasticity. For the regularity of these problems it would be useful to be able to approximate these maps by smooth homeomorphisms in their corresponding Sobolev space (the so-called Ball-Evans problem). We construct a pair of homeomorphisms for which is impossible simultaneously solving the Hajlasz problem. That is we construct a Sobolev homeomorphism equalling identity on the boundary of a cube but with negative Jacobian almost everywhere.