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Zeitschrift für Analysis und ihre Anwendungen

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Volume 34, Issue 3, 2015, pp. 321–342
DOI: 10.4171/ZAA/1542

Published online: 2015-07-08

The Weak Inverse Mapping Theorem

Daniel Campbell[1], Stanislav Hencl[2] and František Konopecký[3]

(1) University of Hradec Králové, Czech Republic
(2) Charles University, Prague, Czech Republic
(3) Charles University, Prague, Czech Republic

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

Keywords: Bilipschitz mappings, inverse mapping theorem

Campbell Daniel, Hencl Stanislav, Konopecký František: The Weak Inverse Mapping Theorem. Z. Anal. Anwend. 34 (2015), 321-342. doi: 10.4171/ZAA/1542