Mappings of finite distortion: Hausdorff measure of zero sets
Let \(f\) be a continuous mapping of the Sobolev class \(W^{1,n}_{\text{loc}}(\Omega,\mathbb{R}^n)\) in a domain \(\Omega\) of \(\mathbb{R}^n\). The mapping \(f\) is said to be quasi-light if for every \(y\) the components of the set \(f^{-1}(y)\) are compact and \(f\) is said to be of finite distortion \(K\) if there is a measurable function \(K: \Omega\to [0,\infty]\) such that \(|Df(x)|^n\leq K(x) J(x,t)\) a.e. If \(f\) satisfies these conditions and if \(K\in L^{n-1}_{\text{loc}}(\Omega)\), then it is shown that \(f\) is discrete and open. This is an extension of the Reshetnyak result and improves a result of \textit{J. Heinonen} and \textit{P. Koskela} [Arch. Ration. Mech. Anal. 125, No. 1, 81-97 (1993; Zbl 0792.30016)] to the borderline exponent \(n-1\) but here the condition on quasi-lightness has been added. The authors also study the Hausdorff dimension of the set \(f^{-1}(y)\) in the above class of mappings under various local integrability conditions for \(K\). The methods rest on the area formula and topological degree formulas for integrals.
scientific article; zbMATH DE number 1889141