JournalsrlmVol. 22, No. 2pp. 207–222

Critical points for Sobolev homeomorphisms

  • Carlo Sbordone

    Università degli Studi di Napoli Federico II, Italy
  • Roberta Schiattarella

    Università degli Studi di Napoli Federico II, Italy
Critical points for Sobolev homeomorphisms cover

Abstract

We consider a class of homeomorphisms f ⁣:ΩR2ontoΩR2f\colon\Omega\subset\mathbb{R}^2\stackrel{\text{onto}}{\longrightarrow} \Omega'\subset\mathbb{R}^2 of the Sobolev space Wloc1,1(Ω;R2)\mathscr {W}^{1,1}_{{\rm loc}}(\Omega; \mathbb{R}^2) whose Jacobian may vanish on a set of positive measure but cannot be zero a.e. in Ω\Omega. This class is defined by the bi-Sobolev condition

\label0f   and   f1Wloc1,1(1)\tag{1}\label{0} f \;\text{ and } \; f^{-1} \in \mathscr {W}^{1,1}_{{\rm loc}}

and reveals useful also in the theory of changes of variables for Sobolev functions.

Cite this article

Carlo Sbordone, Roberta Schiattarella, Critical points for Sobolev homeomorphisms. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 2, pp. 207–222

DOI 10.4171/RLM/596