JournalszaaVol. 34, No. 3pp. 321–342

The Weak Inverse Mapping Theorem

  • Daniel Campbell

    University of Hradec Králové, Czech Republic
  • Stanislav Hencl

    Charles University, Prague, Czech Republic
  • František Konopecký

    Charles University, Prague, Czech Republic
The Weak Inverse Mapping Theorem cover

Abstract

We prove that if a bilipschitz mapping ff is in Wlocm,p(Rn,Rn)W_{\mathrm {loc}}^{m,p}(\mathbb R^n, \mathbb R^n) then the inverse f1f^{-1} is also a Wlocm,pW_{\mathrm {loc}}^{m,p} class mapping. Further we prove that the class of bilipschitz mappings belonging to Wlocm,p(Rn,Rn)W_{\mathrm {loc}}^{m,p} (\mathbb R^n, \mathbb R^n) is closed with respect to composition and multiplication without any restrictions on m,p1m, p \geq 1. These results can be easily extended to smooth nn-dimensional Riemannian manifolds and further we prove a form of the implicit function theorem for Sobolev mappings.

Cite this article

Daniel Campbell, Stanislav Hencl, František Konopecký, The Weak Inverse Mapping Theorem. Z. Anal. Anwend. 34 (2015), no. 3, pp. 321–342

DOI 10.4171/ZAA/1542