Bi-Sobolev mappings and elliptic equations in the plane (Q1018332)

The authors obtain an important generalisation of quasiconformal mappings and their relation to PDEs. Let \(\Omega\), \(\Omega'\) be bounded domains in \(\mathbb{R}^2\). The function \(f=(u,v)\), \(f:\Omega @>\text{onto}>>\Omega'\), is a bi-Sobolev mapping if it is a homeomorphism belonging to the Sobolev space \(W^{1,1}_{\text{loc}} (\Omega;\Omega')\) and its inverse \(f^{-1}\) belongs to \(W^{1,1}_{\text{loc}}(\Omega';\Omega)\). The authors prove that \(u\) and \(v\) have the same critical points; as an application, they show that \(u\) and \(v\) are distributional solutions of the same nontrivial degenerate elliptic equation in divergence form. Whereas to each quasiconformal mapping there corresponds an elliptic PDF, they determine the different situation when \(f=(u,v)\) is only a \(W^{1,1}_{\text{loc}}\)-homeomorphism and the distortion of the mapping is not bounded. They then examine distortion properties of \(W^{1,p}\)-bi-Sobolev mappings in \(\mathbb{R}^n\) and construct a number of useful counterexamples.