Reshetnyak's theorem and the inner distortion (Q652907)

The present paper is devoted to the study of space mappings with finite distortion, intensively investigated last 10-20 years. Let \(D\) be a domain in \({\mathbb R}^n\), \(n\geq 2\). A map \(f:D\rightarrow {\mathbb R}^n\) is said to have finite distortion if \(|Df(x)|^n\leq K(x)J_f(x)\) a.e., where \(|Df(x)|\) is a matrix norm of \(Df(x)\), \(J_f(x)\) denotes a Jacobian of \(f\) at \(x\), and \(K(x)\) is some Lebesgue measurable function \(K:D\rightarrow [1, \infty)\). A mapping \(f:D\rightarrow {\mathbb R}^n\) is called quasilight if the set \(f^{-1}(y)\) is compact for every \(y\in {\mathbb R}^n\). The main result of the paper is the following. Let \(f:D\rightarrow {\mathbb R}^n\) be quasilight with finite distortion. Suppose that \(f\in W_{\text{loc}}^{1, n}(D)\) and that the inner dilatation \(K_I(x)\) of \(f\) at \(x\) is locally integrable in \(D\). Then \(f\) is discrete and open.