Quasiregular mappings of several \(n\)-dimensional variables
Let \(U\) be a domain in \((\mathbb{R}^ n)^ k\) and let \(f=(f^ 1, f^ 2, \dots, f^ m)\) be a mapping from \(U\) to \((\mathbb{R}^ n)^ m\) with Jacobi matrix \(f'(x)\) defined almost everywhere in \(U\). A mapping \(f\) is said to be \(K\)-quasiregular mapping of several \(n\)-dimensional variables if the following inequality holds almost everywhere in \(U\): \[ \sum_{i,j} | f_{ij}'(x) |^ n \leq K \cdot n^{n \over 2} \sum_{i,j} J_{ij} (x,f), \] where \(J_{ij} (x,f) = \text{det} f_{ij}'(x)\) and \(| A | = (\sum_{i,j} a^ 2_{ij})^{1/2}\) for any matrix \(A \in \mathbb{R}^{s \times t}\), \(A = (a_{ij})_{i=1,2, \dots, s;j = 1,2, \dots,t}\). The family of such mappings are denoted by \(Q(K)\). In this paper some properties of such mappings are investigated. In particularly it is proved that for \(K \geq 1\) the family \(Q(K)\) has the following properties. \(1^ \circ\) If \(f_ s:U \to (\mathbb{R}^ n)^ m\), \(s = 1,2, \dots\), is a sequence of mappings in \(Q(K)\) converging to a mapping \(f_ 0\) then \(f_ 0 \in Q(K)\). \(2^ \circ\) Let \(f=(f_ 1, f_ 2, \dots, f_ n) : U \to \mathbb{R}^ n\) be a mapping of the class \(Q(K)\), \((m=1)\), then there are some polynomials of a special type \(Q_ j\), \(P_ j\), \(j=1,2, \dots,n\) such that \[ f_ j(x) = P_ j(x)/Q_ j(x), \quad x \in U, \quad j = 1,2, \dots,n; \] moreover, each \(Q_ j\) nonvanishes in \(U\).
scientific article; zbMATH DE number 652176