Stability of classes of quasiregular mappings in several spatial variables (Q1921756)

Let \((\mathbb{R}^n)^k = \mathbb{R}^n \times \mathbb{R}^n \times \cdots \times \mathbb{R}^n\), \((k\)-times), \(x \in (\mathbb{R}^n)^k\) may be written as \(x = (x_1, x_2, \dots, x_k)\) where \(x_j \in \mathbb{R}^n \), \(j = 1,2, \dots, k\). For a given domain \(U \subset (\mathbb{R}^n)^k\) we introduce a function \(f = (f_1, f_2, \dots, f_m) : U \to (\mathbb{R}^n)^m\) of the class \(W^1_{n, \text{loc}} (U, (\mathbb{R}^n)^m)\) with Jacobi matrix \[ f'(x) = \bigl[ f_{ij}'(x) \bigr]_{i = 1, 2, \dots, m; j = 1, 2, \dots, k} \quad \text{where} \quad \bigl[ f_{ij}' \bigr] = \left[ {\partial f^i \over \partial x_j} \right] \in \mathbb{R}^{n \times n} \] are the matrices of a dimension \(n \times n\). Definition A mapping \(f : U \to (\mathbb{R}^n)^m\) is called the quasiregular mapping of several spatial variables if \(1^\circ\) \(f \in W^1_{n, \text{loc}} (U, (\mathbb{R}^n)^m)\); \(2^\circ\) there exists a constant \(K < \infty\) such that \[ \sum_{i,j} |f_{ij}' |^n \leq K n^{n/2} \cdot \sum_{i,j} \text{det} f_{ij}', \quad \text{where} \quad |A |= \bigl |[a_{rs}] \bigr |= \Bigl( \sum_{r,s} a^2_{rs} \Bigr)^{1/2} \] and let \(\text{det} A\) denote the determinant of \(A\). The constant \(K = K(f)\) is called a coefficient of quasiregularity of a mapping \(f\) and the set of all quasiregular mappings such that \(K(f) \leq K\) is denoted by \(G(K) = G^m_{n,k} (K)\). In this paper some properties of such classes of quasiregular mappings of several spatial variables are investigated. This results in the special cases coincide with the earliest results of the author and the results of Kopylov and Reshetnyak.