The weak inverse mapping theorem (Q496882)

Summary: 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.