BLD-mappings in \(W^{2,2}\) are locally invertible
A mapping \(f\) from a domain \(\Omega\) of \(\mathbb{R}^n\) into \(\mathbb{R}^n\) is called a BLD (= bounded length distortion) mapping if \(f\) is locally uniformly Lipschitz and if the Jacobian determinant \(J(x,f)\) of \(f\) satisfies \(J(x,f) \geq c>0\) a.e. in \(\Omega\), see [\textit{O. Martio}, \textit{J. Väisälä}, Math. Ann. 282, No. 3, 435-443 (1988; Zbl 0632.53021)]. Such a mapping \(f\) needs not to be locally injective in \(\mathbb{R}^n\), \(n\geq 2\), the ordinary winding mapping \((r, \varphi,z) \mapsto(r,2 \varphi,z)\) being a counterexample. This mapping is ``irregular'' on the \(z\)-axis, which is the branch out of \(f\). The authors answer a question of D. Sullivan. Is every BLD mapping \(f\) in the Sobolew space \(W^{2,2}\) a local homeomorphism? This is the optimal result in the BLD-class as a computation of the derivatives of the winding mappings shows. It is rather easy to show that \(W^{2,2+\varepsilon}\) implies local injectivity, but the limiting case requires a careful normal family argument. The argument can also be used for quasiregular mappings in \(\mathbb{R}^n\), \(n\geq 3\).
scientific article; zbMATH DE number 1546887
scientific article